{"id":121166,"date":"2022-05-13T14:55:14","date_gmt":"2022-05-13T09:25:14","guid":{"rendered":"https:\/\/www.mapsofindia.com\/my-india\/?p=121166"},"modified":"2022-05-13T14:55:14","modified_gmt":"2022-05-13T09:25:14","slug":"chapter-1-sets-questions-and-answers-ncert-solutions-for-class-11-maths","status":"publish","type":"post","link":"https:\/\/www.mapsofindia.com\/my-india\/education\/chapter-1-sets-questions-and-answers-ncert-solutions-for-class-11-maths","title":{"rendered":"Chapter 1 – Sets Questions and Answers: NCERT Solutions for Class 11 Maths"},"content":{"rendered":"

Exercise (1.1)<\/h2>\n

1. Which of the following are sets? Justify your answer.
\ni. The collection of all months of a year beginning with the letter J.<\/h2>\n

Ans-
\nTo determine if the given statement is a set
\nA set is a collection of well-defined objects.
\nWe can definitely identify the collection of months beginning with a letter J.
\nThus, the collection of all months of a year beginning with the letter J is the set.<\/h3>\n

ii. The collection of ten most talented writers of India<\/h2>\n

Ans-
\nTo determine if the given statement is a set
\nA set is a collection of well-defined objects.
\nThe criteria for identifying the collection of the ten most talented writers of India may vary from person to person. So it is not a well-defined object.
\nThus, the collection of the ten most talented writers of India is not a set.<\/h3>\n

iii. A team of eleven best cricket batsmen in the world.<\/h2>\n

Ans-
\nTo determine if the given statement is a set
\nA set is a collection of well-defined objects.
\nThe criteria for determining the eleven best cricket batsmen may vary from person to person. So it is not a well-defined object.
\nThus, a team of eleven best cricket batsmen in the world is not a set.<\/h3>\n

iv. The collection of all boys in your class.<\/h2>\n

Ans-
\nTo determine if the given statement is a set
\nA set is a collection of well-defined objects.
\nWe can definitely identify the boys who are all studying in the class. So it is a well-defined object.
\nThus, the collection of all boys in your class is a set.<\/h3>\n

v. The collection of all-natural numbers is less than 100100.<\/h2>\n

Ans-
\nTo determine if the given statement is a set
\nA set is a collection of well-defined objects.
\nWe can identify the natural numbers less than 100100 that can easily be identified. So it is a well-defined object.
\nThus, the collection of all-natural numbers less than 100100 is a set.<\/h3>\n

vi. A collection of novels written by the writer Munshi Prem Chand.<\/h2>\n

Ans-
\nTo determine if the given statement is a set
\nA set is a collection of well-defined objects.
\nWe can identify the books that belong to the writer Munshi Prem Chand. So it is a well-defined object.
\nThus, a collection of novels written by the writer Munshi Prem Chand is a set.<\/h3>\n

vii. The collection of all even integers.<\/h2>\n

Ans-
\nTo determine if the given statement is a set
\nA set is a collection of well-defined objects.
\nWe can identify integers that are all the collection of even integers. So it is not a well-defined object.
\nThus, the collection of all even integers is a set.<\/h3>\n

viii. The collection of questions in this chapter.<\/h2>\n

Ans-
\nTo determine if the given statement is a set
\nA set is a collection of well-defined objects.
\nWe can easily identify the questions that are in this chapter. So it is a well-defined object.
\nThus, the collection of questions in this chapter is a set.<\/h3>\n

ix. A collection of the most dangerous animals in the world.<\/h2>\n

Ans-
\nTo determine if the given statement is a set
\nA set is a collection of well-defined objects.
\nThe criteria for determining the most dangerous animals may vary according to the person. So it is not a well-defined object.
\nThus, the collection of the most dangerous animals in the world is a set.<\/h3>\n

2. Let A={1,2,3,4,5,6}A={1,2,3,4,5,6}. Insert the appropriate symbol \u2208\u2208 or \u2209\u2209 in the blank spaces:
\ni. 5…A5…A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
\nTo insert the appropriate symbol \u2208\u2208 or \u2209\u2209
\nThe number 55 is in the set.
\n\u22345\u2208A\u22345\u2208A<\/h3>\n

ii. 8…A8…A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
\nTo insert the appropriate symbol \u2208\u2208 or \u2209\u2209
\nThe number 88 is not in the set.
\n\u22348\u2209A\u22348\u2209A<\/h3>\n

iii. 0…A0…A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
\nTo insert the appropriate symbol \u2208\u2208 or \u2209\u2209
\nThe number 00 is not in the set.
\n\u22340\u2209A\u22340\u2209A<\/h3>\n

iv. 4…A4…A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
\nTo insert the appropriate symbol \u2208\u2208 or \u2209\u2209
\nThe number 44 is in the set.
\n\u22344\u2208A\u22344\u2208A<\/h3>\n

v. 2…A2…A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
\nTo insert the appropriate symbol \u2208\u2208 or \u2209\u2209
\nThe number 22 is in the set.
\n\u22342\u2208A\u22342\u2208A<\/h3>\n

vi. 10…A10…A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
\nTo insert the appropriate symbol \u2208\u2208 or \u2209\u2209
\nThe number 1010 is not in the set.
\n\u223410\u2209A\u223410\u2209A<\/h3>\n

3. Write the following sets in roster form:
\ni. A={x:xis an integer and -3×7}A={x:xis an integer and -3×7}<\/h2>\n

Ans-
\nGiven that,
\nA={x:xis an integer and -3×7}A={x:xis an integer and -3×7}
\nTo write the above expression in its roaster form
\nIn roaster form, the order in which the elements are listed is immaterial.
\nThe elements of the set are \u22122,\u22121,0,1,2,3,4,5,6\u22122,\u22121,0,1,2,3,4,5,6.
\n\u2234\u2234The roaster form of the set A={x:xis an integer and -3×7}A={x:xis an integer and -3×7} is A={\u22122,\u22121,0,1,2,3,4,5,6}A={\u22122,\u22121,0,1,2,3,4,5,6}.<\/h3>\n

ii. B={x:xis a natural number less than 6}B={x:xis a natural number less than 6}<\/h2>\n

Ans-
\nGiven that,
\nB={x:xis a natural number less than 6}B={x:xis a natural number less than 6}
\nTo write the above expression in its roaster form
\nIn roaster form, the order in which the elements are listed is immaterial.
\nThe elements of the set are 1,2,3,4,51,2,3,4,5.
\n\u2234\u2234The roaster form of the set B={x:xis a natural number less than 6}B={x:xis a natural number less than 6} is B={1,2,3,4,5}B={1,2,3,4,5}.<\/h3>\n

iii. C={x:xis a two-digit natural number such that sum of its digits is 8}C={x:xis a two-digit natural number such that sum of its digits is 8}<\/h2>\n

Ans-
\nGiven that,
\nC={x:xis a two-digit natural number such that sum of its digits is 8}C={x:xis a two-digit natural number such that sum of its digits is 8}
\nTo write the above expression in its roaster form
\nIn roaster form, the order in which the elements are listed is immaterial.
\nThe elements of the set are 17,26,35,44,53,62,71,8017,26,35,44,53,62,71,80 such that their sum is 88
\n\u2234\u2234The roaster form of the set C={x:xis a two-digit natural number such that sum of its digits is 8}C={x:xis a two-digit natural number such that sum of its digits is 8} is {17,26,35,44,53,62,71,80}{17,26,35,44,53,62,71,80}.<\/h3>\n

iv. D={x:xis a prime number which is divisor of 60}D={x:xis a prime number which is divisor of 60}<\/h2>\n

Ans-
\nGiven that,
\nD={x:xis a prime number which is divisor of 60}D={x:xis a prime number which is divisor of 60}
\nTo write the above expression in its roaster form
\nIn roaster form, the order in which the elements are listed is immaterial.
\nThe divisors of 6060 are 2,3,4,5,62,3,4,5,6. Among these, the prime numbers are 2,3,52,3,5
\nThe elements of the set are 2,3,52,3,5.
\n\u2234\u2234The roaster form of the set D={x:xis a prime number which is divisor of 60}D={x:xis a prime number which is divisor of 60} is D={2,3,5}D={2,3,5}.<\/h3>\n

v. E=E= The set of all letters in the word TRIGONOMETRY<\/h2>\n

Ans-
\nGiven that,
\nE=E=The set of all letters in the word TRIGONOMETRY
\nTo write the above expression in its roaster form
\nIn roaster form, the order in which the elements are listed is immaterial.
\nThere are 1212 letters in the word TRIGONOMETRY out of which T, R and O gets repeated.
\nThe elements of the set are T, R, I G, O, N, M, E, Y.
\n\u2234\u2234The roaster form of the set E=E=The set of all letters in the word TRIGONOMETRY is E={T,R,I,G,O,N,M,E,Y}E={T,R,I,G,O,N,M,E,Y}.<\/h3>\n

vi. F=F=The set of all letters in the word BETTER<\/h2>\n

Ans-
\nGiven that,
\nF=F=The set of all letters in the word BTTER
\nTo write the above expression in its roaster form
\nIn roaster form, the order in which the elements are listed is immaterial.
\nThere are 66 letters in the word BETTER out of which E and T are repeated.
\nThe elements of the set are B, E, T, R.
\n\u2234\u2234The roaster form of the set F=F=The set of all letters in the word BTTER
\nis F={B,E,T,R}F={B,E,T,R}.<\/h3>\n

4. Write the following sets in the set builder form:
\ni. (3,6,9,12)(3,6,9,12)<\/h2>\n

Ans-
\nGiven that,
\n{3,6,9,12}{3,6,9,12}
\nTo represent the given set in the set builder form
\nIn set-builder form, all the elements of a set possess a single common property that is not possessed by any element outside the set.
\nFrom the given set, we observe that the numbers in the set are multiple of 33 from 11 to 44 such that {x:x=3n,n\u2208Nand 1\u2264n\u22644}{x:x=3n,n\u2208Nand 1\u2264n\u22644}
\n\u2234{3,6,9,12}={x:x=3n,n\u2208Nand 1\u2264n\u22644}\u2234{3,6,9,12}={x:x=3n,n\u2208Nand 1\u2264n\u22644}<\/h3>\n

ii. {2,4,8,16,32}{2,4,8,16,32}<\/h2>\n

Ans-
\nGiven that,
\n{2,4,8,16,32}{2,4,8,16,32}
\nTo represent the given set in the set builder form
\nIn set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
\nFrom the given set, we observe that the numbers in the set are powers of 22from 11 to 55 such that {x:x=2n,n\u2208Nand 1\u2264n\u22645}{x:x=2n,n\u2208Nand 1\u2264n\u22645}
\n\u2234{2,4,8,16,32}={x:x=2n,n\u2208Nand 1\u2264n\u22645}\u2234{2,4,8,16,32}={x:x=2n,n\u2208Nand 1\u2264n\u22645}<\/h3>\n

iii. {5,25,125,625}{5,25,125,625}<\/h2>\n

Ans-
\nGiven that,
\n{5,25,125,625}{5,25,125,625}
\nTo represent the given set in the set builder form
\nIn set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
\nFrom the given set, we observe that the numbers in the set are powers of 55from 11 to 44 such that {x:x=5n,n\u2208Nand 1\u2264n\u22644}{x:x=5n,n\u2208Nand 1\u2264n\u22644}
\n\u2234{5,25,125,625}={x:x=5n,n\u2208Nand 1\u2264n\u22644}\u2234{5,25,125,625}={x:x=5n,n\u2208Nand 1\u2264n\u22644}<\/h3>\n

iv. {2,4,6,…}{2,4,6,…}<\/h2>\n

Ans-
\nGiven that,
\n{2,4,6,…}{2,4,6,…}
\nTo represent the given set in the set builder form
\nIn set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
\nFrom the given set, we observe that the numbers are the set of all even natural numbers.
\n\u2234{2,4,6,…}={x:xis an even natural number}\u2234{2,4,6,…}={x:xis an even natural number}<\/h3>\n

v) {1,4,9,…100}{1,4,9,…100}<\/h2>\n

Ans-
\nGiven that,
\n{1,4,9,…100}{1,4,9,…100}
\nTo represent the given set in the set builder form
\nIn set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
\nFrom the given set, we observe that the numbers in the set squares of numbers form 11 to 1010 such that {x:x=n2,n\u2208Nand 1\u2264n\u226410}{x:x=n2,n\u2208Nand 1\u2264n\u226410}
\n\u2234{1,4,9,…100}={x:x=n2,n\u2208Nand 1\u2264n\u226410}\u2234{1,4,9,…100}={x:x=n2,n\u2208Nand 1\u2264n\u226410}<\/h3>\n

5. List all the elements of the following sets:
\ni. A={x:xis an odd natural number}A={x:xis an odd natural number}<\/h2>\n

Ans-
\nGiven that,
\nA={x:xis an odd natural number}A={x:xis an odd natural number}
\nTo list the elements of the given set
\nThe odd natural numbers are 1,3,5,…1,3,5,…
\n\u2234\u2234The set A={x:xis an odd natural number}A={x:xis an odd natural number} has the odd natural numbers that are {1,3,5,…}{1,3,5,…}<\/h3>\n

ii. B={x:xis an integer;-12<x<92}B={x:xis an integer;-12<x<92}<\/h2>\n

Ans-
\nGiven that,
\nB={x:xis an integer;-12<x<12}B={x:xis an integer;-12<x<12}
\nTo list the elements of the given set
\n\u221212=\u22120.5\u221212=\u22120.5 and 92=4.592=4.5
\nSo the integers between \u22120.5\u22120.5 and 4.54.5 are 0,1,2,3,40,1,2,3,4
\n\u2234\u2234The set B={x:xis an integer;-12<x<12}B={x:xis an integer;-12<x<12} has an integers that are between {0,1,2,3,4}{0,1,2,3,4}<\/h3>\n

iii. C={x:xis an integer;x2\u22644}C={x:xis an integer;x2\u22644}<\/h2>\n

Ans-
\nGiven that,
\nC={x:xis an integer;x2\u22644}C={x:xis an integer;x2\u22644}
\nTo list the elements of the given set
\nIt is observed that,
\nx2\u22644×2\u22644
\n(\u22122)2=4\u22644(\u22122)2=4\u22644
\n(\u22121)2=1\u22644(\u22121)2=1\u22644
\n(0)2=0\u22644(0)2=0\u22644
\n(1)2=1\u22644(1)2=1\u22644
\n(2)2=4\u22644(2)2=4\u22644
\n\u2234\u2234The set C={x:xis an integer;x2\u22644}C={x:xis an integer;x2\u22644} contains elements such as {\u22122,\u22121,0,1,2}{\u22122,\u22121,0,1,2}<\/h3>\n

iv. D={x:xis a letter in the word ”LOYAL”}D={x:xis a letter in the word ”LOYAL”}<\/h2>\n

Ans-
\nGiven that,
\nD={x:xis a letter in the word ”LOYAL”}D={x:xis a letter in the word ”LOYAL”}
\nTo list the elements of the given set
\nThere are 55 total letters in the given word in which L gets repeated two times.
\nSo the elements in the set are {L,O,Y,A}{L,O,Y,A}
\n\u2234\u2234The set D={x:xis a letter in the word ”LOYAL”}D={x:xis a letter in the word ”LOYAL”} consists the elements {L,O,Y,A}{L,O,Y,A}.<\/h3>\n

v. E={x:xis a month of a year not having 31 days}E={x:xis a month of a year not having 31 days}<\/h2>\n

Ans-
\nGiven that,
\nE={x:xis a month of a year not having 31 days}E={x:xis a month of a year not having 31 days}
\nTo list the elements of the given set
\nThe months that don\u2019t have 3131 are as follows:
\nFebruary, April, June, September, November
\n\u2234\u2234The set E={x:xis a month of a year not having 31 days}E={x:xis a month of a year not having 31 days} consist of the elements such that {February, April, June, September, November}{February, April, June, September, November}<\/h3>\n

vi. F={x:xis a consonant in the English alphabet which precedes k}F={x:xis a consonant in the English alphabet which precedes k}<\/h2>\n

Ans-
\nGiven that,
\nF={x:xis a consonant in the English alphabet which precedes k}F={x:xis a consonant in the English alphabet which precedes k}
\nTo list the elements of the given set
\nThe consonants are letters in English alphabet other than vowels such as a, e, i, o, u and the consonants that precedes k include b, c, d, f, g, h, j
\n\u2234\u2234The set F={x:xis a consonant in the English alphabet which precedes k}F={x:xis a consonant in the English alphabet which precedes k} consists of the set {b,c,d,f,g,h,j}{b,c,d,f,g,h,j}<\/h3>\n

6. Match each of the sets on the left in the roaster form with the same set on the right described in set-builder form.
\ni. {1,2,3,6}{1,2,3,6}<\/h2>\n

Ans-
\nGiven that,
\n{1,2,3,6}{1,2,3,6}
\nTo match the roaster form in the left with the set builder form in the right
\nIn roaster form, the order in which the elements are listed is immaterial.
\nIn set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
\nIt has been observed from the set that these set of numbers are the set of natural numbers which are also the divisors of 66
\nThus, {1,2,3,6}={x:xis a natural number and is a divisor of 6}{1,2,3,6}={x:xis a natural number and is a divisor of 6} is the correct option which is option (c)<\/h3>\n

ii. {2,3}{2,3}<\/h2>\n

Ans-
\nGiven that,
\n{2,3}{2,3}
\nTo match the roaster form in the left with the set builder form in the right
\nIn roaster form, the order in which the elements are listed is immaterial.
\nIn set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
\nIt has been observed from the set that these set of numbers are the set of prime numbers which are also the divisors of 66
\nThus, {2,3}={x:xis a prime number and is a divisor of 6}{2,3}={x:xis a prime number and is a divisor of 6} is the correct option which is option (a)<\/h3>\n

iii. {M,A,T,H,E,I,C,S}{M,A,T,H,E,I,C,S}<\/h2>\n

Ans-
\nGiven that,
\n{M,A,T,H,E,I,C,S}{M,A,T,H,E,I,C,S}
\nTo match the roaster form in the left with the set builder form in the right
\nIn roaster form, the order in which the elements are listed is immaterial.
\nIn set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
\nIt has been observed from the set of these letters of word MATHEMATICS.
\nThus, {M,A,T,H,E,I,C,S}={x:xis a letter of the word MATHEMATICS}{M,A,T,H,E,I,C,S}={x:xis a letter of the word MATHEMATICS} is the correct option which is option (d)<\/h3>\n

iv. {1,3,5,7,9}{1,3,5,7,9}<\/h2>\n

Ans-
\nGiven that,
\n{1,3,5,7,9}{1,3,5,7,9}
\nTo match the roaster form in the left with the set builder form in the right
\nIn roaster form, the order in which the elements are listed is immaterial.
\nIn set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
\nIt has been observed from the set that these sets of numbers are the set of odd numbers that are less than 1010.
\nThus, {1,3,5,7,9}={x:xis a odd number less than 10}{1,3,5,7,9}={x:xis a odd number less than 10} is the correct option which is option (b)<\/h3>\n

Exercise (1.2)<\/h2>\n

1. Which of the following are examples of the null set
\ni. Set of odd natural numbers divisible by 22<\/h2>\n

Ans-
\nGiven that,
\nSet of odd natural numbers divisible by 22
\nTo find if the given statement is an example of null set
\nA set which does not contain any element is called the empty set or the null set or the void set.
\nThere is no odd number that will be divisible by 22 and so this set is a null set.
\n\u2234\u2234The set of odd natural numbers divisible by 22 is a null set.<\/h3>\n

ii. Set of even prime numbers<\/h2>\n

Ans-
\nGiven that,
\nSet of even prime numbers.
\nTo find if the given statement is an example of null set
\nA set which does not contain any element is called the empty set or the null set or the void set.
\nThere was an even number 22, which will be the one and only even prime number. So the set contains an element. So it is not a null set.
\n\u2234\u2234The set of even prime numbers is not a null set.<\/h3>\n

iii. {x:xis a natural numbers, x5 and x7}{x:xis a natural numbers, x5 and x7}<\/h2>\n

Ans-
\nGiven that,
\n{x:xis a natural numbers, x5 and x7}{x:xis a natural numbers, x5 and x7}
\nTo find if the given statement is an example of null set
\nA set which does not contain any element is called the empty set or the null set or the void set.
\nThere was no number that will be less than 55 and greater than 77 simultaneously. So it is a null set
\n\u2234\u2234 {x:xis a natural numbers, x5 and x7}{x:xis a natural numbers, x5 and x7} is a null set<\/h3>\n

iv. {y:yis a point common to any two parallel lines}{y:yis a point common to any two parallel lines}<\/h2>\n

Ans-
\nGiven that,
\n{y:yis a point common to any two parallel lines}{y:yis a point common to any two parallel lines}
\nTo find if the given statement is an example of null set
\nA set which does not contain any element is called the empty set or the null set or the void set.
\nThe parallel lines do not intersect each other. So it does not have a common point of intersection. So it is a null set.
\n\u2234\u2234 {y:yis a point common to any two parallel lines}{y:yis a point common to any two parallel lines}is a null set.<\/h3>\n

2. Which of the following sets are finite or infinite.
\ni. The sets of months of a year<\/h2>\n

Ans-
\nGiven that,
\nThe sets of months of a year
\nTo find if the set is finite or infinite
\nA set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
\nA year has twelve months which has defined number of elements
\n\u2234\u2234The set of months of a year is finite.<\/h3>\n

ii. {1,2,3…}{1,2,3…}<\/h2>\n

Ans-
\nGiven that,
\n{1,2,3…}{1,2,3…}
\nTo find if the set is finite or infinite
\nA set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
\nThe set consists of an infinite number of natural numbers.
\n\u2234\u2234The set {1,2,3…}{1,2,3…} is infinite since it contains an infinite number of elements.<\/h3>\n

iii. {1,2,3,…,99,100}{1,2,3,…,99,100}<\/h2>\n

Ans-
\nGiven that,
\n{1,2,3,…,99,100}{1,2,3,…,99,100}
\nTo find if the set is finite or infinite
\nA set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
\nThis set contains the elements from 11 to 100100which are finite in number.
\n\u2234\u2234The set {1,2,3,…,99,100}{1,2,3,…,99,100} is finite.<\/h3>\n

iv. The set of positive integers greater than 100100<\/h2>\n

Ans-
\nGiven that,
\nThe set of positive integers greater than 100100
\nTo find if the set is finite or infinite
\nA set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
\nThe positive integers which are greater than 100100 are infinite in number.
\n\u2234\u2234The set of positive integers greater than 100100 is infinite.<\/h3>\n

v. The set of prime numbers less than 9999<\/h2>\n

Ans-
\nGiven that,
\nThe set of prime numbers less than 9999
\nTo find if the set is finite or infinite
\nA set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
\nThe prime numbers less than 9999 are finite in number.
\n\u2234\u2234The set of prime numbers less than 9999 is finite.<\/h3>\n

3. State whether each of the following set is finite or infinite:
\ni. The sets of lines which are parallel to xx axis<\/h2>\n

Ans-
\nGiven that,
\nThe set of lines which are parallel to xx axis
\nTo find if the set is finite or infinite
\nA set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
\nThe lines parallel to the xx axis are infinite in number.
\n\u2234\u2234The set of lines parallel to xx axis is infinite.<\/h3>\n

ii. The set of letters in English alphabet<\/h2>\n

Ans-
\nGiven that,
\nThe set of letter sin English alphabet
\nTo find if the set is finite or infinite
\nA set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
\nEnglish alphabet consist of 2626 elements which is finite in number
\n\u2234\u2234The set of letters in the English alphabet is finite.<\/h3>\n

Ans-
\nGiven that,
\nThe set of numbers which are multiple of 55
\nTo find if the set is finite or infinite
\nA set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
\nThe numbers which are all multiple of 55 are infinite in number.
\n\u2234\u2234The set of numbers which are multiple of 55is infinite.<\/h3>\n

iv. The set of animals living on the earth<\/h2>\n

Ans-
\nGiven that,
\nThe set of animals living on the earth
\nTo find if the set is finite or infinite
\nA set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
\nAlthough the number of animals on the earth is quite a big number, it is finite.
\n\u2234\u2234The set of animals living on the earth is finite.<\/h3>\n

v. The set of circles passing through the origin (0,0)(0,0)<\/h2>\n

Ans-
\nGiven that,
\nThe set of circles passing through the origin (0,0)(0,0)
\nTo find if the set is finite or infinite
\nA set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
\nThe number of circles passing through the origin may be infinite in number.
\n\u2234\u2234The set of circles passing through origin (0,0)(0,0) is infinite.<\/h3>\n

4. In the following, state whether A=BA=B or not
\ni. A={a,b,c,d};B={d,c,b,a}A={a,b,c,d};B={d,c,b,a}<\/h2>\n

Ans-
\nGiven that,
\nA={a,b,c,d};B={d,c,b,a}A={a,b,c,d};B={d,c,b,a}
\nTo state whether A=BA=B
\nWe know that the order in which the elements are listed are insignificant. So A=BA=B
\n\u2234A=B\u2234A=B<\/h3>\n

ii. A={4,8,12,16}:B={8,4,16,18}A={4,8,12,16}:B={8,4,16,18}<\/h2>\n

Ans-
\nGiven that,
\nA={4,8,12,16}:B={8,4,16,18}A={4,8,12,16}:B={8,4,16,18}
\nTo state whether A=BA=B
\nWe know that 12\u2208A12\u2208A but 12\u2209B12\u2209B
\n\u2234A\u2260B\u2234A\u2260B<\/h3>\n

iii. A={2,4,6,8,10};B={x:xis a positive integer and x\u226410}A={2,4,6,8,10};B={x:xis a positive integer and x\u226410}
\nAns-<\/h2>\n

Given that,
\nA={2,4,6,8,10};B={x:xis a positive integer and x\u226410}A={2,4,6,8,10};B={x:xis a positive integer and x\u226410}
\nTo state whether A=BA=B
\nA={2,4,6,8,10}A={2,4,6,8,10}
\nThe positive integers less than 1010 are B={2,4,6,8,10}B={2,4,6,8,10} So A=BA=B
\n\u2234A=B\u2234A=B<\/h3>\n

iv. A={x:xis a multiple of 10};B={10,15,20,25,30,…}A={x:xis a multiple of 10};B={10,15,20,25,30,…}<\/h2>\n

Ans-
\nGiven that,
\nA={x:xis a multiple of 10};B={10,15,20,25,30,…}A={x:xis a multiple of 10};B={10,15,20,25,30,…}
\nTo state whether A=BA=B
\nA={10,20,30,40,…}A={10,20,30,40,…}
\nB={10,15,20,25,30,…}B={10,15,20,25,30,…}
\nThe elements of A consists only of multiples of 1010 and not of 55. So A\u2260BA\u2260B
\n\u2234A\u2260B\u2234A\u2260B<\/h3>\n

5. Are the following pair of sets equal? Give reasons.
\ni. A={2,3};B={x:xis solution ofx2+5x+6=0}A={2,3};B={x:xis solution ofx2+5x+6=0}<\/h2>\n

Ans-
\nGiven that,
\nA={2,3};B={x:xis a solution ofx2+5x+6=0}A={2,3};B={x:xis a solution ofx2+5x+6=0}
\nTo state whether A=BA=B
\nSolving x2+5x+6=0x2+5x+6=0,
\nx2+3x+2x+6=0x2+3x+2x+6=0
\n(x+2)(x+3)=0(x+2)(x+3)=0
\nx=\u22122,\u22123x=\u22122,\u22123
\nB={\u22122,\u22123}B={\u22122,\u22123} and A={2,3}A={2,3}
\nSo A\u2260BA\u2260B
\n\u2234A\u2260B\u2234A\u2260B<\/h3>\n

ii. A={x:xis a letter in the word FOLLOW};B={y:yis a letter in the word WOLF}A={x:xis a letter in the word FOLLOW};B={y:yis a letter in the word WOLF}<\/h2>\n

Ans-
\nGiven that,
\nA={x:xis a letter in the word FOLLOW};B={y:yis a letter in the word WOLF}A={x:xis a letter in the word FOLLOW};B={y:yis a letter in the word WOLF}
\nTo state whether A=BA=B
\nA={x:xis a letter in the word FOLLOW}={F,O,L,W}A={x:xis a letter in the word FOLLOW}={F,O,L,W}
\nB={y:yis a letter in the word WOLF}={W,O,L,F}B={y:yis a letter in the word WOLF}={W,O,L,F}
\nWe know that the order in which the elements are listed are insignificant. So A=BA=B
\n\u2234A=B\u2234A=B<\/h3>\n

6. From the sets given below, select equal sets:
\nA={2,4,8,12},B={1,2,3,4},C={4,8,12,14},D={3,1,4,2}A={2,4,8,12},B={1,2,3,4},C={4,8,12,14},D={3,1,4,2} E={\u22121,1},F={0,a},G={1,\u22121},H={0,1}E={\u22121,1},F={0,a},G={1,\u22121},H={0,1}<\/h2>\n

Ans-
\nGiven that,
\nA={2,4,8,12},B={1,2,3,4},C={4,8,12,14},D={3,1,4,2}A={2,4,8,12},B={1,2,3,4},C={4,8,12,14},D={3,1,4,2}
\nE={\u22121,1},F={0,a},G={1,\u22121},H={0,1}E={\u22121,1},F={0,a},G={1,\u22121},H={0,1}
\nTo select equal sets from the given set
\nTwo sets A and B are said to be equal if they have exactly the same elements and we write A = B
\nWe can observe from the sets that,
\n8\u2208A,8\u2209B,8\u2209D,8\u2209E,8\u2209F,8\u2209G,8\u2209H8\u2208A,8\u2209B,8\u2209D,8\u2209E,8\u2209F,8\u2209G,8\u2209H
\nAnd thus
\nA\u2260B,A\u2260D,A\u2260E,A\u2260F,A\u2260G,A\u2260HA\u2260B,A\u2260D,A\u2260E,A\u2260F,A\u2260G,A\u2260H
\nBut 8\u2208C8\u2208C
\nAnd checking other elements,
\n2\u2208A,2\u2209C2\u2208A,2\u2209C
\nSo A\u2260CA\u2260C
\n3\u2208B,3\u2209C,3\u2209E,3\u2209F,3\u2209G,3\u2209H3\u2208B,3\u2209C,3\u2209E,3\u2209F,3\u2209G,3\u2209H
\nAnd thus,
\nB\u2260C,B\u2260E,B\u2260F,B\u2260G,B\u2260HB\u2260C,B\u2260E,B\u2260F,B\u2260G,B\u2260H
\n12\u2208C,12\u2209D,12\u2209E,12\u2209F,12\u2209G,12\u2209H12\u2208C,12\u2209D,12\u2209E,12\u2209F,12\u2209G,12\u2209H
\nAnd thus
\nC\u2260D,C\u2260E,C\u2260F,C\u2260G,C\u2260HC\u2260D,C\u2260E,C\u2260F,C\u2260G,C\u2260H
\n4\u2208D,4\u2209E,4\u2209F,4\u2209G,4\u2209H4\u2208D,4\u2209E,4\u2209F,4\u2209G,4\u2209H
\nAnd thus,
\nD\u2260E,D\u2260F,D\u2260G,D\u2260HD\u2260E,D\u2260F,D\u2260G,D\u2260H
\nSimilarly E\u2260F,E\u2260G,E\u2260HE\u2260F,E\u2260G,E\u2260H
\nF\u2260G,F\u2260HF\u2260G,F\u2260H
\nG\u2260HG\u2260H
\nWe know that the order of the elements I listed are insignificant.
\nSo B=D,E=GB=D,E=G
\n\u2234\u2234He equal sets are B=DB=D and E=GE=G<\/h3>\n

Exercise (1.3)<\/h2>\n

1. Make correct statements by filling in the symbols \u2282\u2282 or \u2282\u0338\u2284 in the blank spaces.
\ni. {2,3,4}…{1,2,3,4,5}{2,3,4}…{1,2,3,4,5}<\/h2>\n

Ans-
\nGiven that,
\n{2,3,4}…{1,2,3,4,5}{2,3,4}…{1,2,3,4,5}
\nTo fill in the correct symbols \u2282\u2282 or \u2282\u0338\u2284 inn the blank spaces
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nThe element in the set {2,3,4}{2,3,4} is also in the set {1,2,3,4,5}{1,2,3,4,5}
\n\u2234{2,3,4}\u2282{1,2,3,4,5}\u2234{2,3,4}\u2282{1,2,3,4,5}<\/h3>\n

ii. {a,b,c}…{b,c,d}{a,b,c}…{b,c,d}<\/h2>\n

Ans-
\nGiven that,
\n{a,b,c}…{b,c,d}{a,b,c}…{b,c,d}
\nTo fill in the correct symbols \u2282\u2282 or \u2282\u0338\u2284 inn the blank spaces
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nThe element in the set {a,b,c}{a,b,c} is not in the set {b,c,d}{b,c,d}
\n\u2234{a,b,c}\u2282\u0338{b,c,d}\u2234{a,b,c}\u2284{b,c,d}<\/h3>\n

iii. {x:xis a student of class XI of your school}…{x:xis a student of class XI of your school}…
\n{x:xis a student of your school}{x:xis a student of your school}<\/h2>\n

Ans-
\nGiven that,
\n{x:xis a student of class XI of your school}…{x:xis a student of class XI of your school}…
\n{x:xis a student of your school}{x:xis a student of your school}
\nTo fill in the correct symbols \u2282\u2282 or \u2282\u0338\u2284 inn the blank spaces
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nThe set of students of class XI would also be inside the set of students in school
\n\u220d{x:xis a student of class XI of your school}\u2282\u220d{x:xis a student of class XI of your school}\u2282 {x:xis a student of your school}{x:xis a student of your school}<\/h3>\n

iv. {x:xis a circle in the plane}…{x:xis a circle in the plane}…
\n{x:xis a circle in the same plane with radius 1 unit}{x:xis a circle in the same plane with radius 1 unit}<\/h2>\n

Ans-
\nGiven that,
\n{x:xis a circle in the plane}…{x:xis a circle in the plane}…
\n{x:xis a circle in the same plane with radius 1 unit}{x:xis a circle in the same plane with radius 1 unit}
\nTo fill in the correct symbols \u2282\u2282 or \u2282\u0338\u2284 inn the blank spaces
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nThe set of circles in the plane with a unit radius will be in the set of the circles in the same plane. So the set of circles in the plane is not in the set of circles with unit radius in the same plane.
\n\u2234{x:xis a circle in the plane}\u2282\u0338\u2234{x:xis a circle in the plane}\u2284 {x:xis a circle in the same plane with radius 1 unit}{x:xis a circle in the same plane with radius 1 unit}<\/h3>\n

v. {x:xis a triangle in the plane}…{x:xis a triangle in the plane}…
\n{x:xis a rectangle in the plane}{x:xis a rectangle in the plane}<\/h2>\n

Ans-
\nGiven that,
\n{x:xis a triangle in the plane}…{x:xis a triangle in the plane}…
\n{x:xis a rectangle in the plane}{x:xis a rectangle in the plane}
\nTo fill in the correct symbols \u2282\u2282 or \u2282\u0338\u2284 inn the blank spaces
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nFrom the given expression itself, we know that the set of triangles in the plane are not in the set of rectangles in the plane.
\n\u2234{x:xis a triangle in the plane}\u2282\u0338\u2234{x:xis a triangle in the plane}\u2284 {x:xis a rectangle in the plane}{x:xis a rectangle in the plane}<\/h3>\n

vi. {x:xis an equilateral triangle in the plane}…{x:xis an equilateral triangle in the plane}…
\n{x:xis a triangle in the plane}{x:xis a triangle in the plane}<\/h2>\n

Ans-
\nGiven that,
\n{x:xis an equilateral triangle in the plane}…{x:xis an equilateral triangle in the plane}… {x:xis a triangle in the plane}{x:xis a triangle in the plane}
\nTo fill in the correct symbols \u2282\u2282 or \u2282\u0338\u2284 inn the blank spaces
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nFrom the above expression, we know that the set of equilateral triangles in the plane is in the set of triangles in the same plane
\n\u2234{x:xis an equilateral triangle in the plane}\u2282\u2234{x:xis an equilateral triangle in the plane}\u2282 {x:xis a triangle in the plane}{x:xis a triangle in the plane}<\/h3>\n

vii. {x:xis an even natural number}…{x:xis an integer}{x:xis an even natural number}…{x:xis an integer}<\/h2>\n

Ans-
\nGiven that,
\n{x:xis an even natural number}…{x:xis an integer}{x:xis an even natural number}…{x:xis an integer}
\nTo fill in the correct symbols \u2282\u2282 or \u2282\u0338\u2284 inn the blank spaces
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nThe set of even natural numbers are in the set of integers.
\n\u2234{x:xis an even natural number}\u2282{x:xis an integer}\u2234{x:xis an even natural number}\u2282{x:xis an integer}<\/h3>\n

2. Examine whether the following statements are true or false
\ni. {a,b}\u2282\u0338{b,c,a}{a,b}\u2284{b,c,a}<\/h2>\n

Ans-
\nGiven that,
\n{a,b}\u2282\u0338{b,c,a}{a,b}\u2284{b,c,a}
\nTo examine whether the above statement is true or false
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nThe element in the set {a,b}{a,b} is also in the set {b,c,a}{b,c,a}
\n\u2234{a,b}\u2282{b,c,a}\u2234{a,b}\u2282{b,c,a}
\n\u2234\u2234The given statement is false<\/h3>\n

ii. {a,e}\u2282{x:xis an vowel in English alpahbet}{a,e}\u2282{x:xis an vowel in English alpahbet}<\/h2>\n

Ans-
\nGiven that,
\n{a,e}\u2282{x:xis an vowel in English alpahbet}{a,e}\u2282{x:xis an vowel in English alpahbet}
\nTo examine whether the above statement is true or false
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nThe element in the set {a,e}{a,e} is also in the set {a,e,i,o,u}{a,e,i,o,u}
\n\u2234{a,e}\u2282{x:xis an vowel in English alpahbet}\u2234{a,e}\u2282{x:xis an vowel in English alpahbet}
\n\u2234\u2234The given statement is true.<\/h3>\n

ii. {1,2,3}\u2282{1,3,5}{1,2,3}\u2282{1,3,5}<\/h2>\n

Ans-
\nGiven that,
\n{1,2,3}\u2282{1,3,5}{1,2,3}\u2282{1,3,5}
\nTo examine whether the above statement is true or false
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nThe element in the set {1,2,3}{1,2,3} is not in the set {1,3,5}{1,3,5} since 2\u2208{1,2,3}2\u2208{1,2,3} and 2\u2209{1,3,5}2\u2209{1,3,5}
\n{1,2,3}\u2282\u0338{1,3,5}{1,2,3}\u2284{1,3,5}
\n\u2234\u2234The given statement is false.<\/h3>\n

iii. {a}\u2282{a,b,c}{a}\u2282{a,b,c}<\/h2>\n

Ans-
\nGiven that,
\n{a}\u2282{a,b,c}{a}\u2282{a,b,c}
\nTo examine whether the above statement is true or false
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nThe element in the set {a}{a} is also in the set {a,b,c}{a,b,c}
\n\u2234{a}\u2282{a,b,c}\u2234{a}\u2282{a,b,c}
\n\u2234\u2234The given statement is true.<\/h3>\n

iv. {a}\u2208{a,b,c}{a}\u2208{a,b,c}<\/h2>\n

Ans-
\nGiven that,
\n{a}\u2208{a,b,c}{a}\u2208{a,b,c}
\nTo examine whether the above statement is true or false
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nThe element in the set {a}{a} and the elements in the set {a,b,c}{a,b,c} are a,b,ca,b,c
\n\u2234{a}\u2282{a,b,c}\u2234{a}\u2282{a,b,c}
\n\u2234\u2234The given statement is false.<\/h3>\n

v. {x:xis an even natural less than 6}\u2282{x:xis an even natural less than 6}\u2282 {x:xis a natural number which divide 36}{x:xis a natural number which divide 36}<\/h2>\n

Ans-
\nGiven that,
\n{x:xis an even natural less than 6}\u2282{x:xis an even natural less than 6}\u2282 {x:xis a natural number which divide 36}{x:xis a natural number which divide 36}To examine whether the above statement is true or false
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\n{x:xis an even natural less than 6}={2,4}{x:xis an even natural less than 6}={2,4}
\n{x:xis a natural number which divide 36}={1,2,3,4,6,9,12,18,36}{x:xis a natural number which divide 36}={1,2,3,4,6,9,12,18,36} \u2234{x:xis an even natural less than 6}\u2282\u2234{x:xis an even natural less than 6}\u2282 {x:xis a natural number which divide 36}{x:xis a natural number which divide 36}
\n\u2234\u2234The given statement is true.<\/h3>\n

3. Let A={1,2,{3,4},5}A={1,2,{3,4},5}. Which of the following statements are incorrect and why?
\ni. {3,4}\u2282A{3,4}\u2282A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if {3,4}\u2282A{3,4}\u2282A is correct or incorrect.
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nFrom the above statement,
\n3\u2208{3,4}3\u2208{3,4}, however 3\u2209A3\u2209A
\n\u2234\u2234The given statement {3,4}\u2282A{3,4}\u2282A is incorrect<\/h3>\n

ii. {3,4}\u2208A{3,4}\u2208A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if {3,4}\u2208A{3,4}\u2208A is correct or incorrect.
\nFrom the above statement,
\n{3,4}{3,4} is an element of A.
\n\u2234{3,4}\u2208A\u2234{3,4}\u2208A
\n\u2234\u2234The given statement is correct.<\/h3>\n

iii. {{3,4}}\u2282A{{3,4}}\u2282A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if {{3,4}}\u2282A{{3,4}}\u2282A is correct or incorrect.
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nFrom the above statement,
\n{3,4}\u2208{{3,4}}{3,4}\u2208{{3,4}} so that {{3,4}}\u2208A{{3,4}}\u2208A
\n\u2234{{3,4}}\u2282A\u2234{{3,4}}\u2282A
\n\u2234\u2234The given statement {{3,4}}\u2282A{{3,4}}\u2282A is correct.<\/h3>\n

iv. 1\u2208A1\u2208A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if 1\u2208A1\u2208A is correct or incorrect.
\nFrom the above statement,
\n11 is an element of A.
\n\u2234\u2234The statement 1\u2208A1\u2208A is a correct statement.<\/h3>\n

v. 1\u2282A1\u2282A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if 1\u2282A1\u2282A is correct or incorrect.
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nFrom the above statement,
\nAn element of a set can never be a subset of itself. So 1\u2282\u0338A1\u2284A
\n\u2234\u2234The given statement 1\u2282A1\u2282A is an incorrect statement.<\/h3>\n

vi. {1,2,5}\u2282A{1,2,5}\u2282A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if {1,2,5}\u2282A{1,2,5}\u2282A is correct or incorrect.
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nFrom the above statement,
\nThe each element of {1,2,5}{1,2,5} is also an element of A, So {1,2,5}\u2282A{1,2,5}\u2282A
\n\u2234\u2234The given statement {1,2,5}\u2282A{1,2,5}\u2282A is a correct statement<\/h3>\n

vii. {1,2,5}\u2208A{1,2,5}\u2208A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if {1,2,5}\u2282A{1,2,5}\u2282A is correct or incorrect.
\nFrom the above statement,
\nElement of {1,2,5}{1,2,5} is not an element of A, So {1,2,5}\u2209A{1,2,5}\u2209A
\nSo the given statement {1,2,5}\u2208A{1,2,5}\u2208A is an incorrect statement.<\/h3>\n

viii. {1,2,3}\u2282A{1,2,3}\u2282A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if {1,2,3}\u2282A{1,2,3}\u2282A is correct or incorrect.
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nFrom the above statement, we notice that,
\n3\u2208{1,2,3}3\u2208{1,2,3}but 3\u2209A3\u2209A
\n{1,2,3}\u2282\u0338A{1,2,3}\u2284A
\n\u2234\u2234The given statement {1,2,3}\u2282A{1,2,3}\u2282A is an incorrect statement.<\/h3>\n

ix. \u2205\u2208A\u2205\u2208A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if \u2205\u2208A\u2205\u2208A is correct or incorrect.
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nFrom the above statement,
\n\u2205\u2205 is not an element of A. So, \u2205\u2209A\u2205\u2209A
\n\u2234\u2234The given statement \u2205\u2208A\u2205\u2208A is an incorrect statement.<\/h3>\n

x. \u2205\u2282A\u2205\u2282A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if \u2205\u2282A\u2205\u2282A is correct or incorrect.
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nFrom the above statement,
\nSince \u2205\u2205 is a subset of every set, \u2205\u2282A\u2205\u2282A
\n\u2234\u2234The given statement \u2205\u2282A\u2205\u2282A is a correct statement.<\/h3>\n

xi. {\u2205}\u2282A{\u2205}\u2282A<\/h2>\n

Ans-
\nGiven that,
\nA={1,2,{3,4},5}A={1,2,{3,4},5}
\nTo find if {\u2205}\u2282A{\u2205}\u2282A is correct or incorrect.
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nFrom the above statement,
\n\u2205\u2205 is an element of A and it is not a subset of A.
\n\u2234\u2234The given statement {\u2205}\u2282A{\u2205}\u2282A is an incorrect statement.<\/h3>\n

3. Write down all the subsets of the following sets:
\ni. {a}{a}<\/h2>\n

Ans-
\nGiven that,
\n{a}{a}
\nTo write the subset of the given sets
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nSubsets of {a}{a} are \u2205\u2205 and {a}{a}<\/h3>\n

ii. {a,b}{a,b}<\/h2>\n

Ans-
\nGiven that,
\n{a,b}{a,b}
\nTo write the subset of the given sets
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nSubsets of {a,b}{a,b} are \u2205\u2205 and {a},{b},{a,b}{a},{b},{a,b}<\/h3>\n

iii. {1,2,3}{1,2,3}<\/h2>\n

Ans-
\nGiven that,
\n{1,2,3}{1,2,3}
\nTo write the subset of the given sets
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nSubsets of {1,2,3}{1,2,3} are \u2205\u2205,{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}<\/h3>\n

iv. \u2205\u2205<\/h2>\n

Ans-
\nGiven that,
\n\u2205\u2205
\nTo write the subset of the given sets
\nA set A is said to be a subset of B if every element of A is also an element of B
\nA\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B
\nSubsets of \u2205\u2205 is \u2205\u2205.<\/h3>\n

4. How many elements has P(A)P(A), if A=\u2205A=\u2205?<\/h2>\n

Ans-
\nGiven that,
\nA=\u2205A=\u2205
\nTo find the number of elements does the P(A)P(A) contain
\nThe collection of all subsets of a set A is called a power set of A and is denoted by P(A)P(A).
\nWe know that if AA is a set with mmelements, that is, n(A)=mn(A)=m, then n[p(A)]=2mn[p(A)]=2m
\nIf A=\u2205A=\u2205 then n(A)=0n(A)=0
\nn[P(A)]=20n[P(A)]=20
\n=1=1
\n\u2234P(A)\u2234P(A) has only one element.<\/h3>\n

5. Write the following as intervals
\ni. {x:x\u2208R,\u22124<x\u22646}{x:x\u2208R,\u22124<x\u22646}<\/h2>\n

Given that,
\n{x:x\u2208R,\u22124<x\u22646}{x:x\u2208R,\u22124<x\u22646}
\nTo write the above expression as intervals
\nThe set of real numbers {y:a<y<b}{y:a<y<b} is called an open interval and is denoted by (a,b)(a,b). The interval which contains the end points also is called close interval and is denoted by[a,b][a,b]
\n\u2234{x:x\u2208R,\u22124<x\u22646}=(\u22124,6]\u2234{x:x\u2208R,\u22124<x\u22646}=(\u22124,6]<\/h3>\n

ii. {x:x\u2208R,\u221212<x<\u221210}{x:x\u2208R,\u221212<x<\u221210}<\/h2>\n

Ans-
\nGiven that,
\n{x:x\u2208R,\u221212<x<\u221210}{x:x\u2208R,\u221212<x<\u221210}
\nTo write the above expression as intervals
\nThe set of real numbers {y:a<y<b}{y:a<y<b} is called an open interval and is denoted by (a,b)(a,b). The interval which contains the end points also is called close interval and is denoted by[a,b][a,b]
\n\u2234{x:x\u2208R,\u221212<x<\u221210}=(\u221212,\u221210)\u2234{x:x\u2208R,\u221212<x<\u221210}=(\u221212,\u221210)<\/h3>\n

iii. {x:x\u2208R,0\u2264x<7}{x:x\u2208R,0\u2264x<7}<\/h2>\n

Ans-
\nGiven that,
\n{x:x\u2208R,0\u2264x<7}{x:x\u2208R,0\u2264x<7}
\nTo write the above expression as intervals
\nThe set of real numbers {y:a<y<b}{y:a<y<b} is called an open interval and is denoted by (a,b)(a,b). The interval which contains the end points also is called close interval and is denoted by[a,b][a,b]
\n\u2235{x:x\u2208R,0\u2264x<7}=[0,7)\u2235{x:x\u2208R,0\u2264x<7}=[0,7)<\/h3>\n

iv. {x:x\u2208R,3\u2264x\u22644}{x:x\u2208R,3\u2264x\u22644}<\/h2>\n

Ans-
\nGiven that,
\n{x:x\u2208R,3\u2264x\u22644}{x:x\u2208R,3\u2264x\u22644}
\nTo write the above expression as intervals
\nThe set of real numbers {y:a<y<b}{y:a<y<b} is called an open interval and is denoted by (a,b)(a,b). The interval which contains the end points also is called close interval and is denoted by[a,b][a,b]
\n\u2234{x:x\u2208R,3\u2264x\u22644}=[3,4]\u2234{x:x\u2208R,3\u2264x\u22644}=[3,4]<\/h3>\n

6. Write the following intervals in set builder form.
\ni. (\u22123,0)(\u22123,0)<\/h2>\n

Ans-
\nGiven that,
\n(\u22123,0)(\u22123,0)
\nTo write the above interval in set builder form
\nThe set of real numbers {y:a<y<b}{y:a<y<b} is called an open interval and is denoted by (a,b)(a,b). The interval which contains the end points also is called close interval and is denoted by[a,b][a,b]
\n\u2234(\u22123,0)={x:x\u2208R,\u22123<x<0}\u2234(\u22123,0)={x:x\u2208R,\u22123<x<0}<\/h3>\n

ii. [6,12][6,12]<\/h2>\n

Ans-
\nGiven that,
\n[6,12][6,12]
\nTo write the above interval in set builder form
\nThe set of real numbers {y:a<y<b}{y:a<y<b} is called an open interval and is denoted by (a,b)(a,b). The interval which contains the end points also is called close interval and is denoted by[a,b][a,b]
\n\u2234[6,12]={x:x\u2208R,6\u2264x\u226412}\u2234[6,12]={x:x\u2208R,6\u2264x\u226412}<\/h3>\n

iii. (6,12](6,12]<\/h2>\n

Ans-
\nGiven that,
\n(6,12](6,12]
\nTo write the above interval in set builder form
\nThe set of real numbers {y:a<y<b}{y:a<y<b} is called an open interval and is denoted by (a,b)(a,b). The interval which contains the end points also is called close interval and is denoted by[a,b][a,b]
\n\u2234(6,12]={x:x\u2208R,6<x\u226412}\u2234(6,12]={x:x\u2208R,6<x\u226412}<\/h3>\n

iv. [\u221223,5)[\u221223,5)<\/h2>\n

Ans-
\nGiven that,
\n[\u221223,5)[\u221223,5)
\nTo write the above interval in set builder form
\nThe set of real numbers {y:a<y<b}{y:a<y<b} is called an open interval and is denoted by (a,b)(a,b). The interval which contains the end points also is called close interval and is denoted by[a,b][a,b]
\n\u2234[\u221223,5)={x:x\u2208R,\u221223\u2264x<5}\u2234[\u221223,5)={x:x\u2208R,\u221223\u2264x<5}<\/h3>\n

7. What universal set(s) would you propose for each of the following:<\/h2>\n

i. The set of right triangles<\/h2>\n

Ans- To propose the universal set for the set of right triangles For the set of right triangles, the universal set can be the set of all kinds of triangles or the set of polygons.<\/h3>\n

ii. The set of isosceles triangles<\/h2>\n

Ans- To propose the universal set for the set of right triangles For the set of isosceles triangles, the universal set can be the set of all kinds of triangles or the set of polygons or the set of two dimensional figures.<\/h3>\n

8. Given the sets A={1,3,5},B={2,4,6}A={1,3,5},B={2,4,6} and C={0,2,4,6,8}C={0,2,4,6,8}, which of the following may be considered as universal set(s) for all the three sets A, B and C?<\/h2>\n

i. {0,1,2,3,4,5,6}{0,1,2,3,4,5,6}<\/h2>\n

Ans- Given that, A={1,3,5},B={2,4,6},C={0,2,4,6,8}A={1,3,5},B={2,4,6},C={0,2,4,6,8} To find if the given set {0,1,2,3,4,5,6}{0,1,2,3,4,5,6} is the universal set of A, B and C It can be observed that, A\u2282A\u2282 {0,1,2,3,4,5,6}{0,1,2,3,4,5,6} B\u2282B\u2282 {0,1,2,3,4,5,6}{0,1,2,3,4,5,6} C\u2282\u0338C\u2284 {0,1,2,3,4,5,6}{0,1,2,3,4,5,6} \u2234\u2234The set {0,1,2,3,4,5,6}{0,1,2,3,4,5,6} cannot be the universal set for the sets A, B and C<\/h3>\n

ii. \u2205\u2205<\/h2>\n

Ans- Given that, A={1,3,5},B={2,4,6},C={0,2,4,6,8}A={1,3,5},B={2,4,6},C={0,2,4,6,8} To find if the given set \u2205\u2205 is the universal set of A, B and C It can be observed that, A\u2282\u0338\u2205A\u2284\u2205 B\u2282\u0338\u2205B\u2284\u2205 C\u2282\u0338\u2205C\u2284\u2205 \u2234\u2234The set \u2205\u2205 cannot be an universal set for A, B and C.<\/h3>\n

iii. {0,1,2,3,4,5,6,7,8,9,10}{0,1,2,3,4,5,6,7,8,9,10}<\/h2>\n

Ans- Given that, A={1,3,5},B={2,4,6},C={0,2,4,6,8}A={1,3,5},B={2,4,6},C={0,2,4,6,8} To find if the given set {0,1,2,3,4,5,6,7,8,9,10}{0,1,2,3,4,5,6,7,8,9,10} is the universal set of A, B and C It can be observe that, A\u2282A\u2282 {0,1,2,3,4,5,6,7,8,9,10}{0,1,2,3,4,5,6,7,8,9,10} B\u2282B\u2282 {0,1,2,3,4,5,6,7,8,9,10}{0,1,2,3,4,5,6,7,8,9,10} C\u2282C\u2282 {0,1,2,3,4,5,6,7,8,9,10}{0,1,2,3,4,5,6,7,8,9,10} \u2234\u2234The set {0,1,2,3,4,5,6,7,8,9,10}{0,1,2,3,4,5,6,7,8,9,10} is the universal set of A, B and C<\/h3>\n

iv. {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8}<\/h2>\n

Ans- Given that, A={1,3,5},B={2,4,6},C={0,2,4,6,8}A={1,3,5},B={2,4,6},C={0,2,4,6,8} To find if the given set {0,1,2,3,4,5,6}{0,1,2,3,4,5,6} is the universal set of A, B and C It can be observed that, A\u2282A\u2282 {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8} B\u2282B\u2282 {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8} C\u2282\u0338C\u2284 {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8} \u2234\u2234The set {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8} is not the universal set of A, B and C<\/h3>\n

Exercise (1.4)<\/h2>\n

1. Find the union of each of following pair of sets<\/h2>\n

i. X={1,3,5},Y={1,2,3}X={1,3,5},Y={1,2,3}<\/h2>\n

Ans- Given that, X={1,3,5},Y={1,2,3}X={1,3,5},Y={1,2,3} To find the union of two sets Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. X\u222aY={1,3,5}\u222a{1,2,3}X\u222aY={1,3,5}\u222a{1,2,3} \u2234X\u222aY={1,2,3,5}\u2234X\u222aY={1,2,3,5}<\/h3>\n

ii. A={a,e,i,o,u},B={a,b,c}A={a,e,i,o,u},B={a,b,c}<\/h2>\n

Ans- Given that, A={a,e,i,o,u},B={a,b,c}A={a,e,i,o,u},B={a,b,c} To find the union of two sets Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. A\u222aB={a,e,i,o,u}\u222a{a,b,c}A\u222aB={a,e,i,o,u}\u222a{a,b,c} \u2234A\u222aB={a,b,c,e,i,o,u}\u2234A\u222aB={a,b,c,e,i,o,u}<\/h3>\n

iii. A={x:xis a natural number an multiple of 3},A={x:xis a natural number an multiple of 3}, B={x:xis a natural number less than 6}B={x:xis a natural number less than 6}<\/h2>\n

Ans- Given that, A={x:xis a natural number an multiple of 3},A={x:xis a natural number an multiple of 3}, B={x:xis a natural number less than 6}B={x:xis a natural number less than 6} To find the union of two sets Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. A={x:xis a natural number an multiple of 3},A={x:xis a natural number an multiple of 3}, ={3,6,9,…}={3,6,9,…} B={x:xis a natural number less than 6}B={x:xis a natural number less than 6} ={1,2,3,4,5,6}={1,2,3,4,5,6} A\u222aB={3,6,9,…}\u222a{1,2,3,4,5,6}A\u222aB={3,6,9,…}\u222a{1,2,3,4,5,6} ={1,2,3,4,5,6,9,12,15…}={1,2,3,4,5,6,9,12,15…} \u2234A\u222aB\u2234A\u222aB ={1,2,3,4,5,6,9,12,15…}={1,2,3,4,5,6,9,12,15…}<\/h3>\n

iv. A={x:xis a natural number 1x\u22646},A={x:xis a natural number 1x\u22646}, B={x:xis a natural number 6×10}B={x:xis a natural number 6×10}<\/h2>\n

Ans- Given that, A={x:xis a natural number 1x\u22646},A={x:xis a natural number 1x\u22646}, B={x:xis a natural number 6×10}B={x:xis a natural number 6×10} To find the union of two sets Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. A={x:xis a natural number 1x\u22646}={2,3,4,5,6}A={x:xis a natural number 1x\u22646}={2,3,4,5,6} B={x:xis a natural number 6×10}={7,8,9}B={x:xis a natural number 6×10}={7,8,9} A\u222aB={2,3,4,5,6}\u222a{7,8,9}A\u222aB={2,3,4,5,6}\u222a{7,8,9} \u2234A\u222aB={2,3,4,5,6,7,8,9}\u2234A\u222aB={2,3,4,5,6,7,8,9}<\/h3>\n

v. A={1,2,3},B=\u2205A={1,2,3},B=\u2205<\/h2>\n

Ans- Given that, A={1,2,3},B=\u2205A={1,2,3},B=\u2205 To find the union of two sets Let A and B be any two sets. The union of A and B is the set that consists of all the elements of A and B. A\u222aB={1,2,3}\u222a\u2205A\u222aB={1,2,3}\u222a\u2205 \u2234A\u222aB={1,2,3}\u2234A\u222aB={1,2,3}<\/h3>\n

2. Let A={a,b}A={a,b} and B={a,b,c}B={a,b,c}. Is A\u2282BA\u2282B? What is A\u222aBA\u222aB?<\/h2>\n

Ans- Given that, A={a,b}A={a,b} and B={a,b,c}B={a,b,c} To find if A\u2282BA\u2282B and A\u222aBA\u222aB A set A is said to be a subset of B if every element of A is also an element of B A\u2282BA\u2282B if a\u2208A,a\u2208Ba\u2208A,a\u2208B It can be observed that A\u2282BA\u2282B A\u222aB={a,b}\u222a{a,b,c}A\u222aB={a,b}\u222a{a,b,c} \u2234A\u222aB={a,b,c}\u2234A\u222aB={a,b,c}<\/h3>\n

3. If A and B are two sets such that A\u2282BA\u2282B then what is A\u22c3BA\u22c3B<\/h2>\n

Ans- Given that, A and B are two sets To find A\u222aBA\u222aB when A\u2282BA\u2282B If A and B are two sets such that A\u2282BA\u2282B, then A\u222aB=BA\u222aB=B<\/h3>\n

4. If A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}A={1,2,3,4},B={3,4,5,6},C={5,6,7,8} and D={7,8,9,10}D={7,8,9,10}; find<\/h2>\n

i. A\u222aBA\u222aB<\/h2>\n

Ans- Given that, A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}, D={7,8,9,10}D={7,8,9,10} To find, A\u222aBA\u222aB Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. A\u222aB={1,2,3,4}\u222a{3,4,5,6}A\u222aB={1,2,3,4}\u222a{3,4,5,6} \u2234A\u222aB={1,2,3,4,5,6}\u2234A\u222aB={1,2,3,4,5,6}<\/h3>\n

ii. A\u222aCA\u222aC<\/h2>\n

Ans- Given that, A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}, D={7,8,9,10}D={7,8,9,10} To find, A\u222aCA\u222aC Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. A\u222aC={1,2,3,4}\u222a{5,6,7,8}A\u222aC={1,2,3,4}\u222a{5,6,7,8} \u2234A\u222aC={1,2,3,4,5,6,7,8}\u2234A\u222aC={1,2,3,4,5,6,7,8}<\/h3>\n

iii. B\u222aCB\u222aC<\/h2>\n

Ans- Given that, A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}, D={7,8,9,10}D={7,8,9,10} To find, B\u222aCB\u222aC Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. B\u222aC={3,4,5,6}\u222a{5,6,7,8}B\u222aC={3,4,5,6}\u222a{5,6,7,8} \u2234B\u222aC={3,4,5,6,7,8}\u2234B\u222aC={3,4,5,6,7,8}<\/h3>\n

iii. B\u222aDB\u222aD<\/h2>\n

Ans- Given that, A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}, D={7,8,9,10}D={7,8,9,10} To find, B\u222aDB\u222aD Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. B\u222aD={3,4,5,6}\u222a{7,8,9,10}B\u222aD={3,4,5,6}\u222a{7,8,9,10} \u2234B\u222aD={3,4,5,6,7,8,9,10}\u2234B\u222aD={3,4,5,6,7,8,9,10}<\/h3>\n

iv. A\u222aB\u222aCA\u222aB\u222aC<\/h2>\n

Ans- Given that, A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}, D={7,8,9,10}D={7,8,9,10} To find, A\u222aB\u222aCA\u222aB\u222aC Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. A\u222aB\u222aC={1,2,3,4}\u222a{3,4,5,6}\u222a{5,6,7,8}A\u222aB\u222aC={1,2,3,4}\u222a{3,4,5,6}\u222a{5,6,7,8} \u2234A\u222aB\u222aC={1,2,3,4,5,6,7,8}\u2234A\u222aB\u222aC={1,2,3,4,5,6,7,8}<\/h3>\n

v. A\u222aB\u222aDA\u222aB\u222aD<\/h2>\n

Ans- Given that, A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}, D={7,8,9,10}D={7,8,9,10} To find, A\u222aB\u222aDA\u222aB\u222aD Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. A\u222aB\u222aD={1,2,3,4}\u222a{3,4,5,6}\u222a{7,8,9,10}A\u222aB\u222aD={1,2,3,4}\u222a{3,4,5,6}\u222a{7,8,9,10} \u2234A\u222aB\u222aD={1,2,3,4,5,6,7,8,9,10}\u2234A\u222aB\u222aD={1,2,3,4,5,6,7,8,9,10}<\/h3>\n

vi. B\u222aC\u222aDB\u222aC\u222aD<\/h2>\n

Ans- Given that, A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}A={1,2,3,4},B={3,4,5,6},C={5,6,7,8}, D={7,8,9,10}D={7,8,9,10} To find, B\u222aC\u222aDB\u222aC\u222aD Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and B. B\u222aC\u222aD={3,4,5,6}\u222a{5,6,7,8}\u222a{7,8,9,10}B\u222aC\u222aD={3,4,5,6}\u222a{5,6,7,8}\u222a{7,8,9,10} \u2234B\u222aC\u222aD={3,4,5,6,7,8,9,10}\u2234B\u222aC\u222aD={3,4,5,6,7,8,9,10}<\/h3>\n

5. Find the intersection of each pair of sets:<\/h2>\n

i. X={1,3,5},Y={1,2,3}X={1,3,5},Y={1,2,3}<\/h2>\n

Ans- Given that, X={1,3,5},Y={1,2,3}X={1,3,5},Y={1,2,3} To find the intersection of the given sets The intersection of sets A and B is the set of all elements which are common to both A and B. X\u2229Y={1,3,5}\u222a{1,2,3}X\u2229Y={1,3,5}\u222a{1,2,3} \u2234X\u2229Y={1,3}\u2234X\u2229Y={1,3}<\/h3>\n

ii. A={a,e,i,o,u},B={a,b,c}A={a,e,i,o,u},B={a,b,c}<\/h2>\n

Ans- Given that, A={a,e,i,o,u},B={a,b,c}A={a,e,i,o,u},B={a,b,c} To find the intersection of the given sets The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229B={a,e,i,o,u}\u222a{a,b,c}A\u2229B={a,e,i,o,u}\u222a{a,b,c} \u2234A\u2229B={a}\u2234A\u2229B={a}<\/h3>\n

iii. A={x:xis a natural number an multiple of 3},A={x:xis a natural number an multiple of 3}, B={x:xis a natural number less than 6}B={x:xis a natural number less than 6}<\/h2>\n

Ans- Given that, A={x:xis a natural number an multiple of 3},A={x:xis a natural number an multiple of 3}, B={x:xis a natural number less than 6}B={x:xis a natural number less than 6} To find the intersection of two sets The intersection of sets A and B is the set of all elements which are common to both A and B. A={x:xis a natural number an multiple of 3},A={x:xis a natural number an multiple of 3}, ={3,6,9,…}={3,6,9,…} B={x:xis a natural number less than 6}B={x:xis a natural number less than 6} ={1,2,3,4,5,6}={1,2,3,4,5,6} A\u2229B={3,6,9,…}\u2229{1,2,3,4,5,6}A\u2229B={3,6,9,…}\u2229{1,2,3,4,5,6} ={3}={3} \u2234A\u2229B\u2234A\u2229B ={3}={3}<\/h3>\n

iv. A={x:xis a natural number 1x\u22646},A={x:xis a natural number 1x\u22646}, B={x:xis a natural number 6×10}B={x:xis a natural number 6×10}<\/h2>\n

Ans- Given that, A={x:xis a natural number 1x\u22646},A={x:xis a natural number 1x\u22646}, B={x:xis a natural number 6×10}B={x:xis a natural number 6×10} To find the intersection of two sets The intersection of sets A and B is the set of all elements which are common to both A and B. A={x:xis a natural number 1x\u22646}={2,3,4,5,6}A={x:xis a natural number 1x\u22646}={2,3,4,5,6} B={x:xis a natural number 6×10}={7,8,9}B={x:xis a natural number 6×10}={7,8,9} A\u2229B={2,3,4,5,6}\u2229{7,8,9}A\u2229B={2,3,4,5,6}\u2229{7,8,9} \u2234A\u2229B=\u2205\u2234A\u2229B=\u2205<\/h3>\n

v. A={1,2,3},B=\u2205A={1,2,3},B=\u2205<\/h2>\n

Ans- Given that, A={1,2,3},B=\u2205A={1,2,3},B=\u2205 To find the intersection of two sets The intersection of sets A and B is the set of all elements which are common to both A and B. A\u22c2B={1,2,3}\u2229\u2205A\u22c2B={1,2,3}\u2229\u2205 \u2234A\u2229B=\u2205\u2234A\u2229B=\u2205<\/h3>\n

6. If A={3,5,7,9,11},B={7,9,11,13},C={11,13,15}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15} and D={15,17}D={15,17}; find<\/h2>\n

i. A\u2229BA\u2229B<\/h2>\n

Ans- Given that, A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17} To find, A\u2229BA\u2229B The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229B={3,5,7,9,11}\u2229{7,9,11,13}A\u2229B={3,5,7,9,11}\u2229{7,9,11,13} \u2234A\u2229B={7,9,11}\u2234A\u2229B={7,9,11}<\/h3>\n

ii. B\u2229CB\u2229C<\/h2>\n

Ans- Given that, A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17} To find, B\u2229CB\u2229C The intersection of sets A and B is the set of all elements which are common to both A and B. B\u2229C={7,9,11,13}\u2229{11,13,15}B\u2229C={7,9,11,13}\u2229{11,13,15} \u2234B\u2229C={11,13}\u2234B\u2229C={11,13}<\/h3>\n

iii. A\u2229C\u2229DA\u2229C\u2229D<\/h2>\n

Ans- Given that, A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17} To find, A\u2229C\u2229DA\u2229C\u2229D The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229C\u2229D={3,5,7,9,11}\u2229{11,13,15}\u2229{15,17}A\u2229C\u2229D={3,5,7,9,11}\u2229{11,13,15}\u2229{15,17} \u2234A\u2229C\u2229D=\u2205\u2234A\u2229C\u2229D=\u2205<\/h3>\n

iv. A\u2229CA\u2229C<\/h2>\n

Ans- Given that, A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17} To find, A\u2229CA\u2229C The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229C={3,5,7,9,11}\u2229{11,13,15}A\u2229C={3,5,7,9,11}\u2229{11,13,15} \u2234A\u2229C={11}\u2234A\u2229C={11}<\/h3>\n

v. B\u2229DB\u2229D<\/h2>\n

Ans- Given that, A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17} To find, B\u2229DB\u2229D The intersection of sets A and B is the set of all elements which are common to both A and B. B\u2229D={7,9,11,13}\u2229{15,17}B\u2229D={7,9,11,13}\u2229{15,17} \u2234B\u2229D=\u2205\u2234B\u2229D=\u2205<\/h3>\n

vi. A\u2229(B\u22c3C)A\u2229(B\u22c3C)<\/h2>\n

Ans- Given that, A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17} To find, A\u2229(B\u222aC)A\u2229(B\u222aC) The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229(B\u222aC)=(A\u2229B)\u222a(A\u2229C)A\u2229(B\u222aC)=(A\u2229B)\u222a(A\u2229C) A\u2229B={3,5,7,9,11}\u2229{7,9,11,13}A\u2229B={3,5,7,9,11}\u2229{7,9,11,13} A\u2229B={7,9,11}A\u2229B={7,9,11} A\u2229D={11}A\u2229D={11} A\u2229(B\u222aC)={7,9,11}\u222a{11}A\u2229(B\u222aC)={7,9,11}\u222a{11} ={11}={11} \u2234A\u2229(B\u222aC)={11}\u2234A\u2229(B\u222aC)={11}<\/h3>\n

vii. A\u2229DA\u2229D<\/h2>\n

Ans- Given that, A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17} To find, A\u2229DA\u2229D The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229D={3,5,7,9,11}\u2229{15,17}A\u2229D={3,5,7,9,11}\u2229{15,17} \u2234A\u2229D=\u2205\u2234A\u2229D=\u2205<\/h3>\n

viii. A\u2229(B\u222aD)A\u2229(B\u222aD)<\/h2>\n

Ans- Given that, A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17} To find, A\u2229(B\u222aD)A\u2229(B\u222aD) The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229(B\u222aD)=(A\u2229B)\u222a(A\u2229D)A\u2229(B\u222aD)=(A\u2229B)\u222a(A\u2229D) A\u2229B={3,5,7,9,11}\u2229{7,9,11,13}A\u2229B={3,5,7,9,11}\u2229{7,9,11,13} A\u2229D=\u2205A\u2229D=\u2205 \u2234A\u2229(B\u222aD)={7,9,11}\u222a\u2205\u2234A\u2229(B\u222aD)={7,9,11}\u222a\u2205 ={7,9,11}={7,9,11}<\/h3>\n

ix. (A\u2229B)\u2229(B\u222aC)(A\u2229B)\u2229(B\u222aC)<\/h2>\n

Ans- Given that, A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17} To find, (A\u2229B)\u2229(B\u222aC)(A\u2229B)\u2229(B\u222aC) The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229B={3,5,7,9,11}\u2229{7,9,11,13}A\u2229B={3,5,7,9,11}\u2229{7,9,11,13} A\u2229B={7,9,11}A\u2229B={7,9,11} B\u222aC={7,9,11,13}\u222a{11,13,15}B\u222aC={7,9,11,13}\u222a{11,13,15} ={7,9,11,13,15}={7,9,11,13,15} (A\u2229B)\u2229(B\u222aC)={7,9,11}\u2229{7,9,11,13,15}(A\u2229B)\u2229(B\u222aC)={7,9,11}\u2229{7,9,11,13,15} ={7,9,11}={7,9,11}<\/h3>\n

x. (A\u222aD)\u2229(B\u222aC)(A\u222aD)\u2229(B\u222aC)<\/h2>\n

Ans- Given that, A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17}A={3,5,7,9,11},B={7,9,11,13},C={11,13,15},D={15,17} To find, (A\u222aD)\u2229(B\u222aC)(A\u222aD)\u2229(B\u222aC) The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229D={3,5,7,9,11}\u2229{15,17}A\u2229D={3,5,7,9,11}\u2229{15,17} A\u2229D={3,5,7,9,11,15,17}A\u2229D={3,5,7,9,11,15,17} B\u222aC={7,9,11,13}\u222a{11,13,15}B\u222aC={7,9,11,13}\u222a{11,13,15} ={7,9,11,13,15}={7,9,11,13,15} (A\u222aD)\u2229(B\u222aC)={3,5,7,9,11,15,17}\u2229{7,9,11,13,15}(A\u222aD)\u2229(B\u222aC)={3,5,7,9,11,15,17}\u2229{7,9,11,13,15} \u2234(A\u222aD)\u2229(B\u222aC)={7,9,11,15}\u2234(A\u222aD)\u2229(B\u222aC)={7,9,11,15}<\/h3>\n

7. If A={x:xis a natural number},B={x:xis an even natural number}A={x:xis a natural number},B={x:xis an even natural number} C={x:xis an odd natural number},D={x:xis a prime number}C={x:xis an odd natural number},D={x:xis a prime number}, find<\/h3>\n

i. A\u2229BA\u2229B<\/h2>\n

Ans- Given that, A={x:xis a natural number}={1,2,3,4,…}A={x:xis a natural number}={1,2,3,4,…} B={x:xis an even natural number}={2,4,6,8…}B={x:xis an even natural number}={2,4,6,8…} C={x:xis an odd natural number}={1,3,5,7,…}C={x:xis an odd natural number}={1,3,5,7,…} D={x:xis a prime number}={2,3,5,7,…}D={x:xis a prime number}={2,3,5,7,…} To find, A\u2229BA\u2229B The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229B={1,2,3,4,…}\u2229{2,4,6,8…}A\u2229B={1,2,3,4,…}\u2229{2,4,6,8…} \u2234A\u2229B=B\u2234A\u2229B=B ={x:xis an even natural number}={x:xis an even natural number}<\/h3>\n

ii. A\u2229CA\u2229C<\/h2>\n

Ans- Given that, A={x:xis a natural number}={1,2,3,4,…}A={x:xis a natural number}={1,2,3,4,…} B={x:xis an even natural number}={2,4,6,8…}B={x:xis an even natural number}={2,4,6,8…} C={x:xis an odd natural number}={1,3,5,7,…}C={x:xis an odd natural number}={1,3,5,7,…} D={x:xis a prime number}={2,3,5,7,…}D={x:xis a prime number}={2,3,5,7,…} To find, A\u2229CA\u2229C The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229C={1,2,3,4,…}\u2229{1,3,5,7…}A\u2229C={1,2,3,4,…}\u2229{1,3,5,7…} \u2234A\u2229C=C\u2234A\u2229C=C {x:xis an odd natural number}{x:xis an odd natural number}<\/h3>\n

iii. A\u2229DA\u2229D<\/h2>\n

Ans- Given that, A={x:xis a natural number}={1,2,3,4,…}A={x:xis a natural number}={1,2,3,4,…} B={x:xis an even natural number}={2,4,6,8…}B={x:xis an even natural number}={2,4,6,8…} C={x:xis an odd natural number}={1,3,5,7,…}C={x:xis an odd natural number}={1,3,5,7,…} D={x:xis a prime number}={2,3,5,7,…}D={x:xis a prime number}={2,3,5,7,…} To find, A\u2229DA\u2229D The intersection of sets A and B is the set of all elements which are common to both A and B. A\u2229D={1,2,3,4,…}\u2229{2,3,5,7,…}A\u2229D={1,2,3,4,…}\u2229{2,3,5,7,…} \u2234A\u2229D=D\u2234A\u2229D=D {x:xis a prime number}{x:xis a prime number}<\/h3>\n

iv. B\u2229CB\u2229C<\/h2>\n

Ans- Given that, A={x:xis a natural number}={1,2,3,4,…}A={x:xis a natural number}={1,2,3,4,…} B={x:xis an even natural number}={2,4,6,8…}B={x:xis an even natural number}={2,4,6,8…} C={x:xis an odd natural number}={1,3,5,7,…}C={x:xis an odd natural number}={1,3,5,7,…} D={x:xis a prime number}={2,3,5,7,…}D={x:xis a prime number}={2,3,5,7,…} To find, B\u2229CB\u2229C The intersection of sets A and B is the set of all elements which are common to both A and B. B\u2229C={2,4,6,8…}\u2229{1,3,5,7…}B\u2229C={2,4,6,8…}\u2229{1,3,5,7…} \u2234B\u2229C=\u2205\u2234B\u2229C=\u2205<\/h3>\n

v. B\u2229DB\u2229D<\/h2>\n

Ans- Given that, A={x:xis a natural number}={1,2,3,4,…}A={x:xis a natural number}={1,2,3,4,…} B={x:xis an even natural number}={2,4,6,8…}B={x:xis an even natural number}={2,4,6,8…} C={x:xis an odd natural number}={1,3,5,7,…}C={x:xis an odd natural number}={1,3,5,7,…} D={x:xis a prime number}={2,3,5,7,…}D={x:xis a prime number}={2,3,5,7,…} To find, B\u2229DB\u2229D The intersection of sets A and B is the set of all elements which are common to both A and B. B\u2229D={2,4,6,8,…}\u2229{2,3,5,7,…}B\u2229D={2,4,6,8,…}\u2229{2,3,5,7,…} \u2234A\u2229D={2}\u2234A\u2229D={2}<\/h3>\n

vi. C\u2229DC\u2229D<\/h2>\n

Ans- Given that, A={x:xis a natural number}={1,2,3,4,…}A={x:xis a natural number}={1,2,3,4,…} B={x:xis an even natural number}={2,4,6,8…}B={x:xis an even natural number}={2,4,6,8…} C={x:xis an odd natural number}={1,3,5,7,…}C={x:xis an odd natural number}={1,3,5,7,…} D={x:xis a prime number}={2,3,5,7,…}D={x:xis a prime number}={2,3,5,7,…} To find, C\u2229DC\u2229D The intersection of sets A and B is the set of all elements which are common to both A and B. C\u2229D={1,3,5,7,…}\u2229{2,3,5,7,…}C\u2229D={1,3,5,7,…}\u2229{2,3,5,7,…} \u2234C\u2229D={x:xis a odd prime number}\u2234C\u2229D={x:xis a odd prime number}<\/h3>\n

.<\/p>\n

8. Which of the following pairs of sets are disjoint
\ni. {1,2,3,4}{1,2,3,4} and {x:xis a antural number and 4\u2264x\u22646}{x:xis a antural number and 4\u2264x\u22646}<\/h2>\n

Ans-
\nGiven that,
\n{1,2,3,4}{1,2,3,4} and
\n{x:xis a natural number and 4\u2264x\u22646}={4,5,6}{x:xis a natural number and 4\u2264x\u22646}={4,5,6}
\nTo find if the given sets are disjoint
\nThe difference between sets A and B in this order is the set of elements that belong to A but not to B.
\n{1,2,3,4}\u2229{4,5,6}={4}{1,2,3,4}\u2229{4,5,6}={4}
\nThus the element exists.
\n\u2234\u2234The given pair of sets is not a disjoint set<\/h3>\n

ii. {a,e,i,o,u}{a,e,i,o,u} and {c,d,e,f}{c,d,e,f}<\/h2>\n

Ans-
\nGiven that,
\n{a,e,i,o,u}{a,e,i,o,u} and {c,d,e,f}{c,d,e,f}
\nTo find if the given sets are disjoint
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\n{a,e,i,o,u}\u2229{c,d,e,f}={e}{a,e,i,o,u}\u2229{c,d,e,f}={e}
\nThus the element exists.
\n\u2234\u2234The given pair of sets is not a disjoint set<\/h3>\n

ii. {x:xis an even integer}{x:xis an even integer} and {x:xis an odd integer}{x:xis an odd integer}<\/h2>\n

Ans-
\nGiven that,
\n{x:xis an even integer}{x:xis an even integer} and
\n{x:xis an odd integer}{x:xis an odd integer}
\nTo find if the given sets are disjoint
\nThe difference between sets A and B in this order is the set of elements that belong to A but not to B.
\n{x:xis an even integer}\u2229{x:xis an even integer}\u2229 {x:xis an odd integer}=\u2205{x:xis an odd integer}=\u2205
\nThus the element does not exist.
\n\u2234\u2234The given pair of sets is a disjoint set<\/h3>\n

9. If A={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20},C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\ni. A\u2212BA\u2212B<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find, A\u2212BA\u2212B
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nA\u2212B={3,6,9,12,15,18,21}\u2212{4,8,12,16,20}A\u2212B={3,6,9,12,15,18,21}\u2212{4,8,12,16,20}
\n\u2234A\u2212B={3,6,9,15,18,21}\u2234A\u2212B={3,6,9,15,18,21}<\/h3>\n

ii. A\u2212CA\u2212C<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find, A\u2212CA\u2212C
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nA\u2212C={3,6,9,12,15,18,21}\u2212{2,4,6,8,10,12,14,16}A\u2212C={3,6,9,12,15,18,21}\u2212{2,4,6,8,10,12,14,16}
\n\u2234A\u2212C={3,9,15,18,21}\u2234A\u2212C={3,9,15,18,21}<\/h3>\n

iii. A\u2212DA\u2212D<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find, A\u2212DA\u2212D
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nA\u2212D={3,6,9,12,15,18,21}\u2212{5,10,15,20}A\u2212D={3,6,9,12,15,18,21}\u2212{5,10,15,20}
\n\u2234A\u2212D={3,6,9,15,18,21}\u2234A\u2212D={3,6,9,15,18,21}<\/h3>\n

iv. B\u2212AB\u2212A<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find, B\u2212AB\u2212A
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nB\u2212A={4,8,12,16,20}\u2212{3,6,9,12,15,18,21}B\u2212A={4,8,12,16,20}\u2212{3,6,9,12,15,18,21}
\n\u2234B\u2212A={4,8,16,20}\u2234B\u2212A={4,8,16,20}<\/h3>\n

v. C\u2212AC\u2212A<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find,C\u2212AC\u2212A
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nC\u2212A={2,4,6,8,10,12,14,16}\u2212{3,6,9,12,15,18,21}C\u2212A={2,4,6,8,10,12,14,16}\u2212{3,6,9,12,15,18,21}
\n\u2234C\u2212A={2,4,8,10,14,16}\u2234C\u2212A={2,4,8,10,14,16}<\/h3>\n

vi. D\u2212AD\u2212A<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find,D\u2212AD\u2212A
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nD\u2212A={5,10,15,20}\u2212{3,6,9,12,15,18,21}D\u2212A={5,10,15,20}\u2212{3,6,9,12,15,18,21}
\n\u2234D\u2212A={5,10,20}\u2234D\u2212A={5,10,20}<\/h3>\n

vii. B\u2212CB\u2212C<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find, B\u2212CB\u2212C
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nB\u2212C={4,8,12,16,20}\u2212{2,4,6,8,10,12,14,16}B\u2212C={4,8,12,16,20}\u2212{2,4,6,8,10,12,14,16}
\n\u2234B\u2212C={20}\u2234B\u2212C={20}<\/h3>\n

viii. B\u2212DB\u2212D<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find, B\u2212DB\u2212D
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nB\u2212D={4,8,12,16,20}\u2212{5,10,15,20}B\u2212D={4,8,12,16,20}\u2212{5,10,15,20}
\n\u2234B\u2212D={4,8,12,16}\u2234B\u2212D={4,8,12,16}<\/h3>\n

ix. C\u2212BC\u2212B<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find, C\u2212BC\u2212B
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nC\u2212B={2,4,6,8,10,12,14,16}\u2212{4,8,12,16,20}C\u2212B={2,4,6,8,10,12,14,16}\u2212{4,8,12,16,20}
\n\u2234C\u2212B={2,6,10,14}\u2234C\u2212B={2,6,10,14}<\/h3>\n

x. D\u2212BD\u2212B<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find, D\u2212BD\u2212B
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nD\u2212B={5,10,15,20}\u2212{4,8,12,16,20}D\u2212B={5,10,15,20}\u2212{4,8,12,16,20}
\n\u2234D\u2212B={5,10,15}\u2234D\u2212B={5,10,15}<\/h3>\n

xi. C\u2212DC\u2212D<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find, C\u2212DC\u2212D
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nC\u2212D={2,4,6,8,10,12,14,16}\u2212{5,10,15,20}C\u2212D={2,4,6,8,10,12,14,16}\u2212{5,10,15,20}
\n\u2234C\u2212D={2,4,6,8,12,14,16}\u2234C\u2212D={2,4,6,8,12,14,16}<\/h3>\n

xii. D\u2212CD\u2212C<\/h2>\n

Ans-
\nGiven that,
\nA={3,6,9,12,15,18,21},B={4,8,12,16,20}A={3,6,9,12,15,18,21},B={4,8,12,16,20} C={2,4,6,8,10,12,14,16},D={5,10,15,20}C={2,4,6,8,10,12,14,16},D={5,10,15,20}
\nTo find, D\u2212CD\u2212C
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nD\u2212C={5,10,15,20}\u2212{2,4,6,8,10,12,14,16}D\u2212C={5,10,15,20}\u2212{2,4,6,8,10,12,14,16}
\n\u2234D\u2212C={5,10,15}\u2234D\u2212C={5,10,15}<\/h3>\n

10. If X={a,b,c,d},Y={f,b,d,g}X={a,b,c,d},Y={f,b,d,g}, find
\ni. X\u2212YX\u2212Y<\/h2>\n

Ans-
\nGiven that,
\nX={a,b,c,d},Y={f,b,d,g}X={a,b,c,d},Y={f,b,d,g}
\nTo find,
\nX\u2212YX\u2212Y
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nX\u2212Y={a,b,c,d}\u2212{f,b,d,g}X\u2212Y={a,b,c,d}\u2212{f,b,d,g}
\n\u2234X\u2212Y={a,c}\u2234X\u2212Y={a,c}<\/h3>\n

ii. Y\u2212XY\u2212X<\/h2>\n

Ans-
\nGiven that,
\nX={a,b,c,d},Y={f,b,d,g}X={a,b,c,d},Y={f,b,d,g}
\nTo find, Y\u2212XY\u2212X
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nY\u2212X={f,b,d,g}\u2212{a,b,c,d}Y\u2212X={f,b,d,g}\u2212{a,b,c,d}
\n\u2234Y\u2212X={f,g}\u2234Y\u2212X={f,g}<\/h3>\n

iii. X\u2229YX\u2229Y<\/h2>\n

Ans-
\nGiven that,
\nX={a,b,c,d},Y={f,b,d,g}X={a,b,c,d},Y={f,b,d,g}
\nTo find,
\nX\u2229YX\u2229Y
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\nX\u2229Y={a,b,c,d}\u2229{f,b,d,g}X\u2229Y={a,b,c,d}\u2229{f,b,d,g}
\n\u2234X\u2229Y={b,d}\u2234X\u2229Y={b,d}<\/h3>\n

11. If R is the real numbers and Q is the set of rational numbers, then what is R\u2212QR\u2212Q?<\/h2>\n

Ans-
\nGiven that,
\nR is the real numbers
\nQ is the set of rational numbers
\nTo find, R\u2212QR\u2212Q
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\n\u2234R\u2212Q\u2234R\u2212Q is the set of irrational numbers.<\/h3>\n

12. State whether each of the following statements is true or false. Justify your answer.
\ni. {2,3,4,5}{2,3,4,5} and {3,6}{3,6} are disjoint sets<\/h2>\n

Ans-
\nGiven that,
\n{2,3,4,5},{3,6}{2,3,4,5},{3,6}
\nTo state whether the given statement is true
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\n{2,3,4,5}\u2229{3,6}={3}{2,3,4,5}\u2229{3,6}={3}
\n\u2234\u2234The given statement is false.<\/h3>\n

ii. {a,e,i,o,u}{a,e,i,o,u} and {a,b,c,d}{a,b,c,d} are disjoint sets<\/h2>\n

Ans-
\nGiven that,
\n{a,e,i,o,u},{a,b,c,d}{a,e,i,o,u},{a,b,c,d}
\nTo state whether the given statement is true
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\n{a,e,i,o,u}\u2229{a,b,c,d}={a}{a,e,i,o,u}\u2229{a,b,c,d}={a}
\n\u2234\u2234The given statement is false.<\/h3>\n

iii. {2,6,10,14}{2,6,10,14} and {3,7,11,15}{3,7,11,15} are disjoint sets<\/h2>\n

Ans-
\nGiven that,
\n{2,6,10,14},{3,7,11,15}{2,6,10,14},{3,7,11,15}
\nTo state whether the given statement is true
\nThe difference of the sets A and B in this order is the set of elements which belong to A but not to B.
\n{2,6,10,14}\u2229{3,7,11,15}=\u2205{2,6,10,14}\u2229{3,7,11,15}=\u2205
\n\u2234\u2234The given statement is true.<\/h3>\n

iv. {2,6,10}{2,6,10} and {3,7,11}{3,7,11} are disjoint sets<\/h2>\n

Ans-
\nGiven that,
\n{2,6,10},{3,7,11}{2,6,10},{3,7,11}
\nTo state whether the given statement is true
\n{2,6,10}\u2229{3,7,11}=\u2205{2,6,10}\u2229{3,7,11}=\u2205
\n\u2234\u2234The given statement is true.<\/h3>\n

Exercise (1.5)<\/h2>\n

1. Let U={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}U={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8} and C={3,4,5,6}C={3,4,5,6}, find
\ni. A\u2032A\u2032<\/h2>\n

Ans-
\nGiven that,
\nU={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}U={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}, C={3,4,5,6}C={3,4,5,6}
\nTo find, A\u2032A\u2032
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\nA\u2032=U\u2212AA\u2032=U\u2212A
\n={1,2,3,4,5,6,7,8,9}\u2212{1,2,3,4}={1,2,3,4,5,6,7,8,9}\u2212{1,2,3,4}
\n={5,6,7,8,9}={5,6,7,8,9}
\n\u2234A\u2032={5,6,7,8,9}\u2234A\u2032={5,6,7,8,9}<\/h3>\n

ii. B\u2032B\u2032<\/h2>\n

Ans-
\nGiven that,
\nU={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}U={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}, C={3,4,5,6}C={3,4,5,6}
\nTo find, B\u2032B\u2032
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\nB\u2032=U\u2212BB\u2032=U\u2212B
\n={1,2,3,4,5,6,7,8,9}\u2212{2,4,6,8}={1,2,3,4,5,6,7,8,9}\u2212{2,4,6,8}
\n={1,3,5,7,9}={1,3,5,7,9}
\n\u2234B\u2032={1,3,5,7,9}\u2234B\u2032={1,3,5,7,9}<\/h3>\n

iii. (A\u222aC)\u2032(A\u222aC)\u2032<\/h2>\n

Ans-
\nGiven that,
\nU={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}U={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}, C={3,4,5,6}C={3,4,5,6}
\nTo find,
\n(A\u222aC)\u2032(A\u222aC)\u2032
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\nA\u222aC={1,2,3,4,5,6}A\u222aC={1,2,3,4,5,6}
\n(A\u222aC)\u2032=U\u2212(A\u222aC)(A\u222aC)\u2032=U\u2212(A\u222aC)
\n={1,2,3,4,5,6,7,8,9}\u2212{1,2,3,4,5,6}={1,2,3,4,5,6,7,8,9}\u2212{1,2,3,4,5,6}
\n={7,8,9}={7,8,9}
\n\u2234(A\u222aC)\u2032={7,8,9}\u2234(A\u222aC)\u2032={7,8,9}<\/h3>\n

iv. (A\u222aB)\u2032(A\u222aB)\u2032<\/h2>\n

Ans-
\nGiven that,
\nU={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}U={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}, C={3,4,5,6}C={3,4,5,6}
\nTo find,
\n(A\u222aB)\u2032(A\u222aB)\u2032
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\nA\u222aB={1,2,3,4,5,6,8}A\u222aB={1,2,3,4,5,6,8}
\n(A\u222aB)\u2032=U\u2212A\u222aB(A\u222aB)\u2032=U\u2212A\u222aB
\n={1,2,3,4,5,6,7,8,9}\u2212{1,2,3,4,5,6,8}={1,2,3,4,5,6,7,8,9}\u2212{1,2,3,4,5,6,8}
\n={5,7,9}={5,7,9}
\n\u2234(A\u222aB)\u2032={5,7,9}\u2234(A\u222aB)\u2032={5,7,9}<\/h3>\n

v. (A\u2032)\u2032(A\u2032)\u2032<\/h2>\n

Ans-
\nGiven that,
\nU={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}U={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}, C={3,4,5,6}C={3,4,5,6}
\nTo find,
\n(A\u2032)\u2032(A\u2032)\u2032
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n(A\u2032)\u2032=A(A\u2032)\u2032=A
\n={1,2,3,4}={1,2,3,4}
\n\u2234(A\u2032)\u2032={1,2,3,4}\u2234(A\u2032)\u2032={1,2,3,4}<\/h3>\n

vi. (B\u2212C)\u2032(B\u2212C)\u2032<\/h2>\n

Ans-
\nGiven that,
\nU={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}U={1,2,3,4,5,6,7,8,9},A={1,2,3,4},B={2,4,6,8}, C={3,4,5,6}C={3,4,5,6}
\nTo find,
\n(B\u2212C)\u2032(B\u2212C)\u2032
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\nB\u2212C={2,8}B\u2212C={2,8}
\n(B\u2212C)\u2032=U\u2212(B\u2212C)(B\u2212C)\u2032=U\u2212(B\u2212C)
\n={1,2,3,4,5,6,7,8,9}\u2212{2,8}={1,2,3,4,5,6,7,8,9}\u2212{2,8}
\n={1,3,4,5,6,7,9}={1,3,4,5,6,7,9}
\n\u2234(B\u2212C)\u2032={1,3,4,5,6,7,9}\u2234(B\u2212C)\u2032={1,3,4,5,6,7,9}<\/h3>\n

2. If U={a,b,c,d,e,f,g,h}U={a,b,c,d,e,f,g,h}, then find the complements of the following sets:
\ni. A={a,b,c}A={a,b,c}<\/h2>\n

Ans-
\nGiven that,
\nU={a,b,c,d,e,f,g,h}U={a,b,c,d,e,f,g,h}
\nA={a,b,c}A={a,b,c}
\nTo find the complement of A
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\nA\u2032=U\u2212AA\u2032=U\u2212A
\n={a,b,c,d,e,f,g,h}\u2212{a,b,c}={a,b,c,d,e,f,g,h}\u2212{a,b,c}
\n={d,e,f,g,h}={d,e,f,g,h}
\n\u2234\u2234The complement of AA is A\u2032={d,e,f,g,h}A\u2032={d,e,f,g,h}<\/h3>\n

ii. B={d,e,f,g}B={d,e,f,g}<\/h2>\n

Ans-
\nGiven that,
\nU={a,b,c,d,e,f,g,h}U={a,b,c,d,e,f,g,h}
\nb={d,e,f,g}b={d,e,f,g}
\nTo find the complement of B
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\nB\u2032=U\u2212BB\u2032=U\u2212B
\n={a,b,c,d,e,f,g,h}\u2212{d,e,f,g}={a,b,c,d,e,f,g,h}\u2212{d,e,f,g}
\n={a,b,c,h}={a,b,c,h}
\n\u2234\u2234The complement of BB is B\u2032={b,e,c,h}B\u2032={b,e,c,h}<\/h3>\n

iii. C={a,c,e,g}C={a,c,e,g}<\/h2>\n

Ans-
\nGiven that,
\nU={a,b,c,d,e,f,g,h}U={a,b,c,d,e,f,g,h}
\nC={a,c,e,g}C={a,c,e,g}
\nTo find the complement of A
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\nC\u2032=U\u2212CC\u2032=U\u2212C
\n={a,b,c,d,e,f,g,h}\u2212{a,c,e,g}={a,b,c,d,e,f,g,h}\u2212{a,c,e,g}
\n={b,d,f,h}={b,d,f,h}
\n\u2234\u2234The complement of CC is C\u2032={b,d,f,h}C\u2032={b,d,f,h}<\/h3>\n

iv. D={f,g,h,a}D={f,g,h,a}<\/h2>\n

Ans-
\nGiven that,
\nU={a,b,c,d,e,f,g,h}U={a,b,c,d,e,f,g,h}
\nA={f,g,h,a}A={f,g,h,a}
\nTo find the complement of A
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\nD\u2032=U\u2212DD\u2032=U\u2212D
\n={a,b,c,d,e,f,g,h}\u2212{f,g,h,a}={a,b,c,d,e,f,g,h}\u2212{f,g,h,a}
\n={b,c,d,e}={b,c,d,e}
\n\u2234\u2234The complement of DD is D\u2032={b,c,d,e}D\u2032={b,c,d,e}<\/h3>\n

3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
\ni. {x:xis an even natural number}{x:xis an even natural number}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural number is the universal set
\nTo find the complement of the set of even natural number
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n\u2234{x:xis an even natural number}\u2032={x:xis an odd natural number}\u2234{x:xis an even natural number}\u2032={x:xis an odd natural number}<\/h3>\n

ii. {x:xis an odd natural number}{x:xis an odd natural number}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural numbers is the universal set
\nTo find the complement of the set of odd natural number
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n\u2234{x:xis an odd natural number}\u2032={x:xis an even natural number}\u2234{x:xis an odd natural number}\u2032={x:xis an even natural number}<\/h3>\n

iii. {x:xis a positive multiple of 3}{x:xis a positive multiple of 3}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural number is the universal set
\nTo find the complement of the set of positive multiples of 33
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n\u2234{x:xis a positive multiple of 3}\u2032={x:x\u2208Nand x is not a positive multiple of 3}\u2234{x:xis a positive multiple of 3}\u2032={x:x\u2208Nand x is not a positive multiple of 3}<\/h3>\n

iv. {x:xis a prime number}{x:xis a prime number}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural number is the universal set
\nTo find the complement of the set of prime number
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n\u2234{x:xis a prime number}\u2032={x:xis a positive composite number and x=1}\u2234{x:xis a prime number}\u2032={x:xis a positive composite number and x=1}<\/h3>\n

v. {x:xis a natural number divisible by 3 and 5}{x:xis a natural number divisible by 3 and 5}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural number is the universal set
\nTo find the complement of the set of natural number divisible by 33 and 55
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n\u2234{x:xis a number divisible by 3 and 5}\u2032=\u2234{x:xis a number divisible by 3 and 5}\u2032=
\n{x:xis a natural number that is not divisible by 3 or 5}{x:xis a natural number that is not divisible by 3 or 5}<\/h3>\n

vi. {x:xis a perfect square}{x:xis a perfect square}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural number is the universal set
\nTo find the complement of the set of perfect squares.
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n\u2234{x:xis a perfect squares}\u2032={x:x\u2208Nand x is not a perfect square}\u2234{x:xis a perfect squares}\u2032={x:x\u2208Nand x is not a perfect square}<\/h3>\n

vii. {x:xis a perfect cube}{x:xis a perfect cube}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural number is the universal set
\nTo find the complement of the set of perfect cube
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n\u2234{x:xis a perfect cube}\u2032={x:x\u2208Nand x is not a perfect cube}\u2234{x:xis a perfect cube}\u2032={x:x\u2208Nand x is not a perfect cube}<\/h3>\n

viii. {x:x+5=8}{x:x+5=8}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural number is the universal set
\nTo find the complement of {x:x+5=8}{x:x+5=8}
\nx+5=8x+5=8
\nx=3x=3
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n\u2234{x:x+5=8}\u2032={x:x\u2208Nand x\u22603}\u2234{x:x+5=8}\u2032={x:x\u2208Nand x\u22603}<\/h3>\n

ix. {x:2x+5=9}{x:2x+5=9}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural number is the universal set
\nTo find the complement of the
\n{x:2x+5=9}{x:2x+5=9}
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n2x+5=92x+5=9
\n2x=42x=4
\nx=2x=2
\n\u2234{x:2x+5=9}\u2032={x:x\u2208Nand x\u22602}\u2234{x:2x+5=9}\u2032={x:x\u2208Nand x\u22602}<\/h3>\n

x. {x:x\u22657}{x:x\u22657}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural number is the universal set
\nTo find the complement of
\n{x:x\u22657}{x:x\u22657}
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n\u2234{x:x\u22657}\u2032={x:x\u2208Nand x7}\u2234{x:x\u22657}\u2032={x:x\u2208Nand x7}<\/h3>\n

xi. {x:x\u2208Nand 2x+110}{x:x\u2208Nand 2x+110}<\/h2>\n

Ans-
\nGiven that,
\nThe set of natural number is the universal set
\nTo find the complement of the
\n{x:x\u2208Nand 2x+110}{x:x\u2208Nand 2x+110}
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\n2x+1>102x+1>10
\n2x>92x>9
\nx>92x>92
\n\u2234{x:x\u2208Nand 2x+110}\u2032\u2234{x:x\u2208Nand 2x+110}\u2032 ={x:x\u2208Nand x\u226492}={x:x\u2208Nand x\u226492}<\/h3>\n

4. If U={1,2,3,4,5,6,7,8,9},A={2,4,6,8}U={1,2,3,4,5,6,7,8,9},A={2,4,6,8} and B={2,3,5,7}B={2,3,5,7}. Verify that,
\ni. (A\u222aB)\u2032=A\u2032\u2229B\u2032(A\u222aB)\u2032=A\u2032\u2229B\u2032<\/h2>\n

Ans-
\nGiven that.
\nU={1,2,3,4,5,6,7,8,9}U={1,2,3,4,5,6,7,8,9}
\nA={2,4,6,8}A={2,4,6,8}
\nB={2,3,5,7}B={2,3,5,7}
\nTo prove that (A\u222aB)\u2032=A\u2032\u2229B\u2032(A\u222aB)\u2032=A\u2032\u2229B\u2032
\nA\u222aB={2,4,6,8}\u222a{2,3,5,7}A\u222aB={2,4,6,8}\u222a{2,3,5,7}
\n={2,3,4,5,6,7,8}={2,3,4,5,6,7,8}
\n(A\u222aB)\u2032=U=A\u222aB(A\u222aB)\u2032=U=A\u222aB
\n={1,9}={1,9}
\nA\u2032=U\u2212AA\u2032=U\u2212A
\n={1,3,5,7,9}={1,3,5,7,9}
\nB\u2032=U\u2212BB\u2032=U\u2212B
\n={1,4,6,8,9}={1,4,6,8,9}
\nA\u2032\u2229B\u2032={1,3,5,7,9}\u2229{1,4,6,8,9}A\u2032\u2229B\u2032={1,3,5,7,9}\u2229{1,4,6,8,9}
\n={1,9}={1,9}
\nHence it has been proved that (A\u222aB)\u2032=A\u2032\u2229B\u2032(A\u222aB)\u2032=A\u2032\u2229B\u2032<\/h3>\n

ii. (A\u2229B)\u2032=A\u2032\u222aB\u2032(A\u2229B)\u2032=A\u2032\u222aB\u2032<\/h2>\n

Ans-
\nGiven that.
\nU={1,2,3,4,5,6,7,8,9}U={1,2,3,4,5,6,7,8,9}
\nA={2,4,6,8}A={2,4,6,8}
\nB={2,3,5,7}B={2,3,5,7}
\nTo prove that (A\u2229B)\u2032=A\u2032\u222aB\u2032(A\u2229B)\u2032=A\u2032\u222aB\u2032
\nA\u2229B={2,4,6,8}\u2229{2,3,5,7}A\u2229B={2,4,6,8}\u2229{2,3,5,7}
\n={2}={2}
\n(A\u2229B)\u2032=U\u2212A\u2229B(A\u2229B)\u2032=U\u2212A\u2229B
\n={1,3,4,5,6,7,8,9}={1,3,4,5,6,7,8,9}
\nA\u2032=U\u2212AA\u2032=U\u2212A
\n={1,3,5,7,9}={1,3,5,7,9}
\nB\u2032=U\u2212BB\u2032=U\u2212B
\n={1,4,6,8,9}={1,4,6,8,9}
\nA\u2032\u222aB\u2032={1,3,5,7,9}\u222a{1,4,6,8,9}A\u2032\u222aB\u2032={1,3,5,7,9}\u222a{1,4,6,8,9}
\n={1,3,4,5,6,7,8,9}={1,3,4,5,6,7,8,9}
\nHence it has been proved that (A\u2229B)\u2032=A\u2032\u222aB\u2032(A\u2229B)\u2032=A\u2032\u222aB\u2032<\/h3>\n

5. Draw appropriate Venn diagrams for each of the following:
\ni. (A\u222aB)\u2032(A\u222aB)\u2032<\/h2>\n

Ans-
\nTo draw the Venn diagram for (A\u222aB)\u2032(A\u222aB)\u2032
\n)<\/h3>\n

ii. A\u2032\u2229B\u2032A\u2032\u2229B\u2032<\/h2>\n

Ans-
\nTo draw the Venn diagram for A\u2032\u2229B\u2032A\u2032\u2229B\u2032
\n)<\/h3>\n

iii. (A\u2229B)\u2032(A\u2229B)\u2032<\/h2>\n

Ans-
\nTo draw the Venn diagram for (A\u2229B)\u2032(A\u2229B)\u2032<\/h3>\n

iv. A\u2032\u222aB\u2032A\u2032\u222aB\u2032<\/h2>\n

Ans-
\nTo draw the Venn diagram for A\u2032\u222aB\u2032A\u2032\u222aB\u2032
\n)<\/h3>\n

6. Let UU be the set of all triangles in a plane. If AA is the set of all triangles with at least one angle different from 60\u221860\u2218, what is A\u2032A\u2032?<\/h2>\n

Ans-
\nGiven that,
\nUU is the set of all triangles in the plane
\nA=A=Set of triangles different form 60\u221860\u2218
\nTo find A\u2032A\u2032
\nThe complement of set A is the set of all elements of U which are not the elements of A.
\nA\u2032=U\u2212AA\u2032=U\u2212A
\n==Set of all equilateral triangles
\n\u2234A\u2032\u2234A\u2032 is the set of all equilateral triangles<\/h3>\n

7. Fill in the blanks to make each of the following a true statement:
\ni. A\u222aA\u2032=…A\u222aA\u2032=…<\/h2>\n

Ans-
\nTo fill the blanks given in the statement
\nThe union of the set and its complement is the universal set
\n\u2234A\u222aA\u2032=U\u2234A\u222aA\u2032=U<\/h3>\n

ii. \u2205\u2032\u2229A=…\u2205\u2032\u2229A=…<\/h2>\n

Ans-
\nTo fill the blanks given in the statement
\nWe know that,
\n\u2205\u2032\u2229A=U\u2229A=A\u2205\u2032\u2229A=U\u2229A=A
\n\u2234\u2205\u2032\u2229A=A\u2234\u2205\u2032\u2229A=A<\/h3>\n

iii. A\u2229A\u2032=…A\u2229A\u2032=…<\/h2>\n

Ans-
\nTo fill the blanks given in the statement
\nThe intersection of the set and its complement is an empty set.
\n\u2234A\u2229A\u2032=\u2205\u2234A\u2229A\u2032=\u2205<\/h3>\n

iv. U\u2032\u2229A=…U\u2032\u2229A=…<\/h2>\n

Ans-
\nTo fill the blanks given in the statement
\nWe know that,
\n\u2205\u2229A=U\u2032\u2229A=\u2205\u2205\u2229A=U\u2032\u2229A=\u2205
\n\u2234U\u2032\u2229A=\u2205\u2234U\u2032\u2229A=\u2205<\/h3>\n

Exercise (1.6)<\/h2>\n

1. If XX and YY are two sets such that n(X)=17,n(Y)=23n(X)=17,n(Y)=23 and n(X\u222aY)=38n(X\u222aY)=38, find n(X\u2229Y)n(X\u2229Y)<\/h2>\n

Ans-
\nGiven that,
\nn(X)=17n(X)=17
\nn(Y)=23n(Y)=23
\nn(X\u222aY)=38n(X\u222aY)=38
\nTo find,
\nn(X\u2229Y)n(X\u2229Y)
\nWe know that,
\nn(X\u222aY)=n(X)+n(Y)\u2212n(X\u2229Y)n(X\u222aY)=n(X)+n(Y)\u2212n(X\u2229Y)
\nn(X\u2229Y)=(17+23)\u221238n(X\u2229Y)=(17+23)\u221238
\n\u2234n(X\u2229Y)=2\u2234n(X\u2229Y)=2<\/h3>\n

2. If XX and YYare two sets such that X\u222aYX\u222aYhas1818 elements, XX has 88 elements and YY has 1515 elements: how many elements does X\u2229YX\u2229Y have?<\/h2>\n

Ans-
\nGiven that,
\nn(X)=8n(X)=8
\nn(Y)=15n(Y)=15
\nn(X\u222aY)=18n(X\u222aY)=18
\nTo find,
\nn(X\u2229Y)n(X\u2229Y)
\nWe know that,
\nn(X\u222aY)=n(X)+n(Y)\u2212n(X\u2229Y)n(X\u222aY)=n(X)+n(Y)\u2212n(X\u2229Y)
\nn(X\u2229Y)=(8+15)\u221218n(X\u2229Y)=(8+15)\u221218
\n\u2234n(X\u2229Y)=5\u2234n(X\u2229Y)=5<\/h3>\n

3. In a group of 400400 people, 250250can speak Hindi and 200200 can speak English. How many people can speak both Hindi and English?<\/h2>\n

Ans-
\nLet the set of people who are speaking Hindi are denoted by HH and the set of people who are speaking English be denoted by EE
\nGiven that,
\nGiven that,
\nn(H)=250n(H)=250
\nn(E)=200n(E)=200
\nn(H\u222aE)=400n(H\u222aE)=400
\nTo find,
\nn(H\u2229E)n(H\u2229E)
\nWe know that,
\nn(H\u222aE)=n(H)+n(E)\u2212n(H\u2229E)n(H\u222aE)=n(H)+n(E)\u2212n(H\u2229E)
\nn(H\u2229E)=(250+200)\u2212400n(H\u2229E)=(250+200)\u2212400
\nn(H\u2229E)=50n(H\u2229E)=50
\n\u223450\u223450 people can speak both English and Hindi.<\/h3>\n

4. If SS and TT are two sets such that SS has 2121 elements, TT has 3232 elements, and S\u2229TS\u2229T has 1111 elements, how many elements does S\u222aTS\u222aT have?<\/h2>\n

Ans-
\nGiven that,
\nn(S)=21n(S)=21
\nn(T)=32n(T)=32
\nn(S\u2229T)=11n(S\u2229T)=11
\nTo find,
\nn(S\u222aT)n(S\u222aT)
\nWe know that,
\nn(S\u222aT)=n(S)+n(T)\u2212n(S\u2229T)n(S\u222aT)=n(S)+n(T)\u2212n(S\u2229T)
\nn(S\u222aT)=21+32\u221211n(S\u222aT)=21+32\u221211
\n\u2234n(S\u222aT)=42\u2234n(S\u222aT)=42
\n\u2234S\u222aT\u2234S\u222aT have 4242 elements<\/h3>\n

5. If XX and YYare two sets such that XX has4040 elements and X\u222aYX\u222aY has 6060 elements, X\u2229YX\u2229Y have 1010 elements, how many elements does YY have?<\/h2>\n

Ans-
\nGiven that,
\nn(X)=40n(X)=40
\nn(X\u222aY)=60n(X\u222aY)=60
\nn(X\u2229Y)=10n(X\u2229Y)=10
\nTo find,
\nn(Y)n(Y)
\nWe know that,
\nn(X\u222aY)=n(X)+n(Y)\u2212n(X\u2229Y)n(X\u222aY)=n(X)+n(Y)\u2212n(X\u2229Y)
\n60=40+n(Y)\u22121060=40+n(Y)\u221210
\nn(Y)=60\u2212(40\u221210)n(Y)=60\u2212(40\u221210)
\n=30=30
\n\u2234n(Y)\u2234n(Y) has 3030 elements.<\/h3>\n

6. In a group of 7070 people, 3737 like coffee, 5252 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?<\/h2>\n

Ans-
\nLet the people who like coffee be denoted by CC and the set of people who like tea be denoted by TT
\nGiven that,
\nn(C)=37n(C)=37
\nn(T)=52n(T)=52
\nn(C\u222aT)=70n(C\u222aT)=70
\nTo find,
\nn(C\u2229T)n(C\u2229T)
\nWe know that,
\nn(C\u222aT)=n(C)+n(T)\u2212n(C\u2229T)n(C\u222aT)=n(C)+n(T)\u2212n(C\u2229T)
\n70=37+52\u2212n(C\u2229T)70=37+52\u2212n(C\u2229T)
\nn(C\u2229T)=(37+52)\u221270n(C\u2229T)=(37+52)\u221270
\n\u2234n(C\u2229T)=19\u2234n(C\u2229T)=19
\n\u2234\u2234The number of people who like both coffee and tea are 1919 people<\/h3>\n

7. In a group of 6565 people, 4040 like cricket, 1010 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?<\/h2>\n

Ans-
\nLet the number of people who like cricket be denoted by CC and the number of people who like tennis be denoted by TT
\nGiven that,
\nn(C)=40n(C)=40
\nn(C\u222aT)=65n(C\u222aT)=65
\nn(C\u2229T)=10n(C\u2229T)=10
\nTo find,
\nn(T)n(T)
\nn(T\u2212C)n(T\u2212C)
\nWe know that,
\nn(C\u222aT)=n(C)+n(T)\u2212n(C\u2229T)n(C\u222aT)=n(C)+n(T)\u2212n(C\u2229T)
\n65=40+n(T)\u22121065=40+n(T)\u221210
\nn(T)=65\u221230n(T)=65\u221230
\n\u2234n(T)=35\u2234n(T)=35
\nThe number of people who like Tennis is 3535 people.
\nNow, (T\u2212C)\u222a(T\u2229C)=T(T\u2212C)\u222a(T\u2229C)=T
\nAnd (T\u2212C)\u2229(T\u2229C)=\u2205(T\u2212C)\u2229(T\u2229C)=\u2205
\nn(T)=n(T\u2212C)+n(T\u2229C)n(T)=n(T\u2212C)+n(T\u2229C)
\n35=n(T\u2212C)+1035=n(T\u2212C)+10
\nn(T\u2212C)=25n(T\u2212C)=25
\n\u2234\u2234The number of people who like only Tennis is 2525 people.<\/h3>\n

8. In a committee 5050 people speak French, 2020 people speak Spanish and 1010 people speak both Spanish and French. How many speak at least one of these two languages?<\/h2>\n

Ans-
\nLet the people who speak French be denoted by FF and the number of people who speak Spanish be denoted by SS
\nGiven that,
\nn(S)=20n(S)=20
\nn(F)=50n(F)=50
\nn(S\u2229F)=10n(S\u2229F)=10
\nTo find,
\nn(S\u222aF)n(S\u222aF)
\nWe know that,
\nn(S\u222aF)=n(S)+n(F)\u2212n(S\u2229F)n(S\u222aF)=n(S)+n(F)\u2212n(S\u2229F)
\n=20+50\u221210=20+50\u221210
\n=60=60
\n\u2234\u2234The number of people who can speak at least one of the languages is 6060 people.<\/h3>\n","protected":false},"excerpt":{"rendered":"

Exercise (1.1) 1. Which of the following are sets? Justify your answer. i. The collection of all months of a year beginning with the letter J. Ans- To determine if the given statement is a set A set is a collection of well-defined objects. We can definitely identify the collection of months beginning with a […]<\/p>\n","protected":false},"author":21830,"featured_media":121167,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":{"0":"post-121166","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-education"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/121166","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/users\/21830"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/comments?post=121166"}],"version-history":[{"count":1,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/121166\/revisions"}],"predecessor-version":[{"id":121180,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/121166\/revisions\/121180"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media\/121167"}],"wp:attachment":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media?parent=121166"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/categories?post=121166"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/tags?post=121166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}