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 letter J.
Thus, the collection of all months of a year beginning with the letter J is the set.
ii. The collection of ten most talented writers of India
Ans-
To determine if the given statement is a set
A set is a collection of well-defined objects.
The 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.
Thus, the collection of the ten most talented writers of India is not a set.
iii. A team of eleven best cricket batsmen in the world.
Ans-
To determine if the given statement is a set
A set is a collection of well-defined objects.
The criteria for determining the eleven best cricket batsmen may vary from person to person. So it is not a well-defined object.
Thus, a team of eleven best cricket batsmen in the world is not a set.
iv. The collection of all boys in your class.
Ans-
To determine if the given statement is a set
A set is a collection of well-defined objects.
We can definitely identify the boys who are all studying in the class. So it is a well-defined object.
Thus, the collection of all boys in your class is a set.
v. The collection of all-natural numbers is less than 100100.
Ans-
To determine if the given statement is a set
A set is a collection of well-defined objects.
We can identify the natural numbers less than 100100 that can easily be identified. So it is a well-defined object.
Thus, the collection of all-natural numbers less than 100100 is a set.
vi. A collection of novels written by the writer Munshi Prem Chand.
Ans-
To determine if the given statement is a set
A set is a collection of well-defined objects.
We can identify the books that belong to the writer Munshi Prem Chand. So it is a well-defined object.
Thus, a collection of novels written by the writer Munshi Prem Chand is a set.
vii. The collection of all even integers.
Ans-
To determine if the given statement is a set
A set is a collection of well-defined objects.
We can identify integers that are all the collection of even integers. So it is not a well-defined object.
Thus, the collection of all even integers is a set.
viii. The collection of questions in this chapter.
Ans-
To determine if the given statement is a set
A set is a collection of well-defined objects.
We can easily identify the questions that are in this chapter. So it is a well-defined object.
Thus, the collection of questions in this chapter is a set.
ix. A collection of the most dangerous animals in the world.
Ans-
To determine if the given statement is a set
A set is a collection of well-defined objects.
The criteria for determining the most dangerous animals may vary according to the person. So it is not a well-defined object.
Thus, the collection of the most dangerous animals in the world is a set.
2. Let A={1,2,3,4,5,6}A={1,2,3,4,5,6}. Insert the appropriate symbol ∈∈ or ∉∉ in the blank spaces:
i. 5…A5…A
Ans-
Given that,
A={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
To insert the appropriate symbol ∈∈ or ∉∉
The number 55 is in the set.
∴5∈A∴5∈A
ii. 8…A8…A
Ans-
Given that,
A={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
To insert the appropriate symbol ∈∈ or ∉∉
The number 88 is not in the set.
∴8∉A∴8∉A
iii. 0…A0…A
Ans-
Given that,
A={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
To insert the appropriate symbol ∈∈ or ∉∉
The number 00 is not in the set.
∴0∉A∴0∉A
iv. 4…A4…A
Ans-
Given that,
A={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
To insert the appropriate symbol ∈∈ or ∉∉
The number 44 is in the set.
∴4∈A∴4∈A
v. 2…A2…A
Ans-
Given that,
A={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
To insert the appropriate symbol ∈∈ or ∉∉
The number 22 is in the set.
∴2∈A∴2∈A
vi. 10…A10…A
Ans-
Given that,
A={1,2,3,4,5,6,}A={1,2,3,4,5,6,}
To insert the appropriate symbol ∈∈ or ∉∉
The number 1010 is not in the set.
∴10∉A∴10∉A
3. Write the following sets in roster form:
i. A={x:xis an integer and -3×7}A={x:xis an integer and -3×7}
Ans-
Given that,
A={x:xis an integer and -3×7}A={x:xis an integer and -3×7}
To write the above expression in its roaster form
In roaster form, the order in which the elements are listed is immaterial.
The elements of the set are −2,−1,0,1,2,3,4,5,6−2,−1,0,1,2,3,4,5,6.
∴∴The roaster form of the set A={x:xis an integer and -3×7}A={x:xis an integer and -3×7} is A={−2,−1,0,1,2,3,4,5,6}A={−2,−1,0,1,2,3,4,5,6}.
ii. B={x:xis a natural number less than 6}B={x:xis a natural number less than 6}
Ans-
Given that,
B={x:xis a natural number less than 6}B={x:xis a natural number less than 6}
To write the above expression in its roaster form
In roaster form, the order in which the elements are listed is immaterial.
The elements of the set are 1,2,3,4,51,2,3,4,5.
∴∴The 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}.
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}
Ans-
Given that,
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}
To write the above expression in its roaster form
In roaster form, the order in which the elements are listed is immaterial.
The 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
∴∴The 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}.
iv. D={x:xis a prime number which is divisor of 60}D={x:xis a prime number which is divisor of 60}
Ans-
Given that,
D={x:xis a prime number which is divisor of 60}D={x:xis a prime number which is divisor of 60}
To write the above expression in its roaster form
In roaster form, the order in which the elements are listed is immaterial.
The divisors of 6060 are 2,3,4,5,62,3,4,5,6. Among these, the prime numbers are 2,3,52,3,5
The elements of the set are 2,3,52,3,5.
∴∴The 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}.
v. E=E= The set of all letters in the word TRIGONOMETRY
Ans-
Given that,
E=E=The set of all letters in the word TRIGONOMETRY
To write the above expression in its roaster form
In roaster form, the order in which the elements are listed is immaterial.
There are 1212 letters in the word TRIGONOMETRY out of which T, R and O gets repeated.
The elements of the set are T, R, I G, O, N, M, E, Y.
∴∴The 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}.
vi. F=F=The set of all letters in the word BETTER
Ans-
Given that,
F=F=The set of all letters in the word BTTER
To write the above expression in its roaster form
In roaster form, the order in which the elements are listed is immaterial.
There are 66 letters in the word BETTER out of which E and T are repeated.
The elements of the set are B, E, T, R.
∴∴The roaster form of the set F=F=The set of all letters in the word BTTER
is F={B,E,T,R}F={B,E,T,R}.
4. Write the following sets in the set builder form:
i. (3,6,9,12)(3,6,9,12)
Ans-
Given that,
{3,6,9,12}{3,6,9,12}
To represent the given set in the set builder form
In set-builder form, all the elements of a set possess a single common property that is not possessed by any element outside the set.
From 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∈Nand 1≤n≤4}{x:x=3n,n∈Nand 1≤n≤4}
∴{3,6,9,12}={x:x=3n,n∈Nand 1≤n≤4}∴{3,6,9,12}={x:x=3n,n∈Nand 1≤n≤4}
ii. {2,4,8,16,32}{2,4,8,16,32}
Ans-
Given that,
{2,4,8,16,32}{2,4,8,16,32}
To represent the given set in the set builder form
In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
From the given set, we observe that the numbers in the set are powers of 22from 11 to 55 such that {x:x=2n,n∈Nand 1≤n≤5}{x:x=2n,n∈Nand 1≤n≤5}
∴{2,4,8,16,32}={x:x=2n,n∈Nand 1≤n≤5}∴{2,4,8,16,32}={x:x=2n,n∈Nand 1≤n≤5}
iii. {5,25,125,625}{5,25,125,625}
Ans-
Given that,
{5,25,125,625}{5,25,125,625}
To represent the given set in the set builder form
In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
From the given set, we observe that the numbers in the set are powers of 55from 11 to 44 such that {x:x=5n,n∈Nand 1≤n≤4}{x:x=5n,n∈Nand 1≤n≤4}
∴{5,25,125,625}={x:x=5n,n∈Nand 1≤n≤4}∴{5,25,125,625}={x:x=5n,n∈Nand 1≤n≤4}
iv. {2,4,6,…}{2,4,6,…}
Ans-
Given that,
{2,4,6,…}{2,4,6,…}
To represent the given set in the set builder form
In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
From the given set, we observe that the numbers are the set of all even natural numbers.
∴{2,4,6,…}={x:xis an even natural number}∴{2,4,6,…}={x:xis an even natural number}
v) {1,4,9,…100}{1,4,9,…100}
Ans-
Given that,
{1,4,9,…100}{1,4,9,…100}
To represent the given set in the set builder form
In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
From the given set, we observe that the numbers in the set squares of numbers form 11 to 1010 such that {x:x=n2,n∈Nand 1≤n≤10}{x:x=n2,n∈Nand 1≤n≤10}
∴{1,4,9,…100}={x:x=n2,n∈Nand 1≤n≤10}∴{1,4,9,…100}={x:x=n2,n∈Nand 1≤n≤10}
5. List all the elements of the following sets:
i. A={x:xis an odd natural number}A={x:xis an odd natural number}
Ans-
Given that,
A={x:xis an odd natural number}A={x:xis an odd natural number}
To list the elements of the given set
The odd natural numbers are 1,3,5,…1,3,5,…
∴∴The 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,…}
ii. B={x:xis an integer;-12<x<92}B={x:xis an integer;-12<x<92}
Ans-
Given that,
B={x:xis an integer;-12<x<12}B={x:xis an integer;-12<x<12}
To list the elements of the given set
−12=−0.5−12=−0.5 and 92=4.592=4.5
So the integers between −0.5−0.5 and 4.54.5 are 0,1,2,3,40,1,2,3,4
∴∴The 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}
iii. C={x:xis an integer;x2≤4}C={x:xis an integer;x2≤4}
Ans-
Given that,
C={x:xis an integer;x2≤4}C={x:xis an integer;x2≤4}
To list the elements of the given set
It is observed that,
x2≤4×2≤4
(−2)2=4≤4(−2)2=4≤4
(−1)2=1≤4(−1)2=1≤4
(0)2=0≤4(0)2=0≤4
(1)2=1≤4(1)2=1≤4
(2)2=4≤4(2)2=4≤4
∴∴The set C={x:xis an integer;x2≤4}C={x:xis an integer;x2≤4} contains elements such as {−2,−1,0,1,2}{−2,−1,0,1,2}
iv. D={x:xis a letter in the word ”LOYAL”}D={x:xis a letter in the word ”LOYAL”}
Ans-
Given that,
D={x:xis a letter in the word ”LOYAL”}D={x:xis a letter in the word ”LOYAL”}
To list the elements of the given set
There are 55 total letters in the given word in which L gets repeated two times.
So the elements in the set are {L,O,Y,A}{L,O,Y,A}
∴∴The 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}.
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}
Ans-
Given that,
E={x:xis a month of a year not having 31 days}E={x:xis a month of a year not having 31 days}
To list the elements of the given set
The months that don’t have 3131 are as follows:
February, April, June, September, November
∴∴The 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}
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}
Ans-
Given that,
F={x:xis a consonant in the English alphabet which precedes k}F={x:xis a consonant in the English alphabet which precedes k}
To list the elements of the given set
The 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
∴∴The 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}
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.
i. {1,2,3,6}{1,2,3,6}
Ans-
Given that,
{1,2,3,6}{1,2,3,6}
To match the roaster form in the left with the set builder form in the right
In roaster form, the order in which the elements are listed is immaterial.
In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
It has been observed from the set that these set of numbers are the set of natural numbers which are also the divisors of 66
Thus, {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)
ii. {2,3}{2,3}
Ans-
Given that,
{2,3}{2,3}
To match the roaster form in the left with the set builder form in the right
In roaster form, the order in which the elements are listed is immaterial.
In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
It has been observed from the set that these set of numbers are the set of prime numbers which are also the divisors of 66
Thus, {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)
iii. {M,A,T,H,E,I,C,S}{M,A,T,H,E,I,C,S}
Ans-
Given that,
{M,A,T,H,E,I,C,S}{M,A,T,H,E,I,C,S}
To match the roaster form in the left with the set builder form in the right
In roaster form, the order in which the elements are listed is immaterial.
In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
It has been observed from the set of these letters of word MATHEMATICS.
Thus, {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)
iv. {1,3,5,7,9}{1,3,5,7,9}
Ans-
Given that,
{1,3,5,7,9}{1,3,5,7,9}
To match the roaster form in the left with the set builder form in the right
In roaster form, the order in which the elements are listed is immaterial.
In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
It has been observed from the set that these sets of numbers are the set of odd numbers that are less than 1010.
Thus, {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)
Exercise (1.2)
1. Which of the following are examples of the null set
i. Set of odd natural numbers divisible by 22
Ans-
Given that,
Set of odd natural numbers divisible by 22
To find if the given statement is an example of null set
A set which does not contain any element is called the empty set or the null set or the void set.
There is no odd number that will be divisible by 22 and so this set is a null set.
∴∴The set of odd natural numbers divisible by 22 is a null set.
ii. Set of even prime numbers
Ans-
Given that,
Set of even prime numbers.
To find if the given statement is an example of null set
A set which does not contain any element is called the empty set or the null set or the void set.
There 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.
∴∴The set of even prime numbers is not a null set.
iii. {x:xis a natural numbers, x5 and x7}{x:xis a natural numbers, x5 and x7}
Ans-
Given that,
{x:xis a natural numbers, x5 and x7}{x:xis a natural numbers, x5 and x7}
To find if the given statement is an example of null set
A set which does not contain any element is called the empty set or the null set or the void set.
There was no number that will be less than 55 and greater than 77 simultaneously. So it is a null set
∴∴ {x:xis a natural numbers, x5 and x7}{x:xis a natural numbers, x5 and x7} is a null set
iv. {y:yis a point common to any two parallel lines}{y:yis a point common to any two parallel lines}
Ans-
Given that,
{y:yis a point common to any two parallel lines}{y:yis a point common to any two parallel lines}
To find if the given statement is an example of null set
A set which does not contain any element is called the empty set or the null set or the void set.
The parallel lines do not intersect each other. So it does not have a common point of intersection. So it is a null set.
∴∴ {y:yis a point common to any two parallel lines}{y:yis a point common to any two parallel lines}is a null set.
2. Which of the following sets are finite or infinite.
i. The sets of months of a year
Ans-
Given that,
The sets of months of a year
To find if the set is finite or infinite
A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
A year has twelve months which has defined number of elements
∴∴The set of months of a year is finite.
ii. {1,2,3…}{1,2,3…}
Ans-
Given that,
{1,2,3…}{1,2,3…}
To find if the set is finite or infinite
A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
The set consists of an infinite number of natural numbers.
∴∴The set {1,2,3…}{1,2,3…} is infinite since it contains an infinite number of elements.
iii. {1,2,3,…,99,100}{1,2,3,…,99,100}
Ans-
Given that,
{1,2,3,…,99,100}{1,2,3,…,99,100}
To find if the set is finite or infinite
A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
This set contains the elements from 11 to 100100which are finite in number.
∴∴The set {1,2,3,…,99,100}{1,2,3,…,99,100} is finite.
iv. The set of positive integers greater than 100100
Ans-
Given that,
The set of positive integers greater than 100100
To find if the set is finite or infinite
A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
The positive integers which are greater than 100100 are infinite in number.
∴∴The set of positive integers greater than 100100 is infinite.
v. The set of prime numbers less than 9999
Ans-
Given that,
The set of prime numbers less than 9999
To find if the set is finite or infinite
A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
The prime numbers less than 9999 are finite in number.
∴∴The set of prime numbers less than 9999 is finite.
3. State whether each of the following set is finite or infinite:
i. The sets of lines which are parallel to xx axis
Ans-
Given that,
The set of lines which are parallel to xx axis
To find if the set is finite or infinite
A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
The lines parallel to the xx axis are infinite in number.
∴∴The set of lines parallel to xx axis is infinite.
ii. The set of letters in English alphabet
Ans-
Given that,
The set of letter sin English alphabet
To find if the set is finite or infinite
A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
English alphabet consist of 2626 elements which is finite in number
∴∴The set of letters in the English alphabet is finite.
Ans-
Given that,
The set of numbers which are multiple of 55
To find if the set is finite or infinite
A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
The numbers which are all multiple of 55 are infinite in number.
∴∴The set of numbers which are multiple of 55is infinite.
iv. The set of animals living on the earth
Ans-
Given that,
The set of animals living on the earth
To find if the set is finite or infinite
A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
Although the number of animals on the earth is quite a big number, it is finite.
∴∴The set of animals living on the earth is finite.
v. The set of circles passing through the origin (0,0)(0,0)
Ans-
Given that,
The set of circles passing through the origin (0,0)(0,0)
To find if the set is finite or infinite
A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.
The number of circles passing through the origin may be infinite in number.
∴∴The set of circles passing through origin (0,0)(0,0) is infinite.
4. In the following, state whether A=BA=B or not
i. A={a,b,c,d};B={d,c,b,a}A={a,b,c,d};B={d,c,b,a}
Ans-
Given that,
A={a,b,c,d};B={d,c,b,a}A={a,b,c,d};B={d,c,b,a}
To state whether A=BA=B
We know that the order in which the elements are listed are insignificant. So A=BA=B
∴A=B∴A=B
ii. A={4,8,12,16}:B={8,4,16,18}A={4,8,12,16}:B={8,4,16,18}
Ans-
Given that,
A={4,8,12,16}:B={8,4,16,18}A={4,8,12,16}:B={8,4,16,18}
To state whether A=BA=B
We know that 12∈A12∈A but 12∉B12∉B
∴A≠B∴A≠B
iii. A={2,4,6,8,10};B={x:xis a positive integer and x≤10}A={2,4,6,8,10};B={x:xis a positive integer and x≤10}
Ans-
Given that,
A={2,4,6,8,10};B={x:xis a positive integer and x≤10}A={2,4,6,8,10};B={x:xis a positive integer and x≤10}
To state whether A=BA=B
A={2,4,6,8,10}A={2,4,6,8,10}
The positive integers less than 1010 are B={2,4,6,8,10}B={2,4,6,8,10} So A=BA=B
∴A=B∴A=B
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,…}
Ans-
Given that,
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,…}
To state whether A=BA=B
A={10,20,30,40,…}A={10,20,30,40,…}
B={10,15,20,25,30,…}B={10,15,20,25,30,…}
The elements of A consists only of multiples of 1010 and not of 55. So A≠BA≠B
∴A≠B∴A≠B
5. Are the following pair of sets equal? Give reasons.
i. A={2,3};B={x:xis solution ofx2+5x+6=0}A={2,3};B={x:xis solution ofx2+5x+6=0}
Ans-
Given that,
A={2,3};B={x:xis a solution ofx2+5x+6=0}A={2,3};B={x:xis a solution ofx2+5x+6=0}
To state whether A=BA=B
Solving x2+5x+6=0x2+5x+6=0,
x2+3x+2x+6=0x2+3x+2x+6=0
(x+2)(x+3)=0(x+2)(x+3)=0
x=−2,−3x=−2,−3
B={−2,−3}B={−2,−3} and A={2,3}A={2,3}
So A≠BA≠B
∴A≠B∴A≠B
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}
Ans-
Given that,
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}
To state whether A=BA=B
A={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}
B={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}
We know that the order in which the elements are listed are insignificant. So A=BA=B
∴A=B∴A=B
6. From the sets given below, select equal sets:
A={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={−1,1},F={0,a},G={1,−1},H={0,1}E={−1,1},F={0,a},G={1,−1},H={0,1}
Ans-
Given that,
A={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={−1,1},F={0,a},G={1,−1},H={0,1}E={−1,1},F={0,a},G={1,−1},H={0,1}
To select equal sets from the given set
Two sets A and B are said to be equal if they have exactly the same elements and we write A = B
We can observe from the sets that,
8∈A,8∉B,8∉D,8∉E,8∉F,8∉G,8∉H8∈A,8∉B,8∉D,8∉E,8∉F,8∉G,8∉H
And thus
A≠B,A≠D,A≠E,A≠F,A≠G,A≠HA≠B,A≠D,A≠E,A≠F,A≠G,A≠H
But 8∈C8∈C
And checking other elements,
2∈A,2∉C2∈A,2∉C
So A≠CA≠C
3∈B,3∉C,3∉E,3∉F,3∉G,3∉H3∈B,3∉C,3∉E,3∉F,3∉G,3∉H
And thus,
B≠C,B≠E,B≠F,B≠G,B≠HB≠C,B≠E,B≠F,B≠G,B≠H
12∈C,12∉D,12∉E,12∉F,12∉G,12∉H12∈C,12∉D,12∉E,12∉F,12∉G,12∉H
And thus
C≠D,C≠E,C≠F,C≠G,C≠HC≠D,C≠E,C≠F,C≠G,C≠H
4∈D,4∉E,4∉F,4∉G,4∉H4∈D,4∉E,4∉F,4∉G,4∉H
And thus,
D≠E,D≠F,D≠G,D≠HD≠E,D≠F,D≠G,D≠H
Similarly E≠F,E≠G,E≠HE≠F,E≠G,E≠H
F≠G,F≠HF≠G,F≠H
G≠HG≠H
We know that the order of the elements I listed are insignificant.
So B=D,E=GB=D,E=G
∴∴He equal sets are B=DB=D and E=GE=G
Exercise (1.3)
1. Make correct statements by filling in the symbols ⊂⊂ or ⊄⊄ in the blank spaces.
i. {2,3,4}…{1,2,3,4,5}{2,3,4}…{1,2,3,4,5}
Ans-
Given that,
{2,3,4}…{1,2,3,4,5}{2,3,4}…{1,2,3,4,5}
To fill in the correct symbols ⊂⊂ or ⊄⊄ inn the blank spaces
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
The 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}
∴{2,3,4}⊂{1,2,3,4,5}∴{2,3,4}⊂{1,2,3,4,5}
ii. {a,b,c}…{b,c,d}{a,b,c}…{b,c,d}
Ans-
Given that,
{a,b,c}…{b,c,d}{a,b,c}…{b,c,d}
To fill in the correct symbols ⊂⊂ or ⊄⊄ inn the blank spaces
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
The element in the set {a,b,c}{a,b,c} is not in the set {b,c,d}{b,c,d}
∴{a,b,c}⊄{b,c,d}∴{a,b,c}⊄{b,c,d}
iii. {x:xis a student of class XI of your school}…{x:xis a student of class XI of your school}…
{x:xis a student of your school}{x:xis a student of your school}
Ans-
Given that,
{x:xis a student of class XI of your school}…{x:xis a student of class XI of your school}…
{x:xis a student of your school}{x:xis a student of your school}
To fill in the correct symbols ⊂⊂ or ⊄⊄ inn the blank spaces
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
The set of students of class XI would also be inside the set of students in school
∍{x:xis a student of class XI of your school}⊂∍{x:xis a student of class XI of your school}⊂ {x:xis a student of your school}{x:xis a student of your school}
iv. {x:xis a circle in the plane}…{x:xis a circle in the plane}…
{x:xis a circle in the same plane with radius 1 unit}{x:xis a circle in the same plane with radius 1 unit}
Ans-
Given that,
{x:xis a circle in the plane}…{x:xis a circle in the plane}…
{x:xis a circle in the same plane with radius 1 unit}{x:xis a circle in the same plane with radius 1 unit}
To fill in the correct symbols ⊂⊂ or ⊄⊄ inn the blank spaces
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
The 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.
∴{x:xis a circle in the plane}⊄∴{x:xis a circle in the plane}⊄ {x:xis a circle in the same plane with radius 1 unit}{x:xis a circle in the same plane with radius 1 unit}
v. {x:xis a triangle in the plane}…{x:xis a triangle in the plane}…
{x:xis a rectangle in the plane}{x:xis a rectangle in the plane}
Ans-
Given that,
{x:xis a triangle in the plane}…{x:xis a triangle in the plane}…
{x:xis a rectangle in the plane}{x:xis a rectangle in the plane}
To fill in the correct symbols ⊂⊂ or ⊄⊄ inn the blank spaces
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
From the given expression itself, we know that the set of triangles in the plane are not in the set of rectangles in the plane.
∴{x:xis a triangle in the plane}⊄∴{x:xis a triangle in the plane}⊄ {x:xis a rectangle in the plane}{x:xis a rectangle in the plane}
vi. {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}
Ans-
Given that,
{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}
To fill in the correct symbols ⊂⊂ or ⊄⊄ inn the blank spaces
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
From the above expression, we know that the set of equilateral triangles in the plane is in the set of triangles in the same plane
∴{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}
vii. {x:xis an even natural number}…{x:xis an integer}{x:xis an even natural number}…{x:xis an integer}
Ans-
Given that,
{x:xis an even natural number}…{x:xis an integer}{x:xis an even natural number}…{x:xis an integer}
To fill in the correct symbols ⊂⊂ or ⊄⊄ inn the blank spaces
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
The set of even natural numbers are in the set of integers.
∴{x:xis an even natural number}⊂{x:xis an integer}∴{x:xis an even natural number}⊂{x:xis an integer}
2. Examine whether the following statements are true or false
i. {a,b}⊄{b,c,a}{a,b}⊄{b,c,a}
Ans-
Given that,
{a,b}⊄{b,c,a}{a,b}⊄{b,c,a}
To examine whether the above statement is true or false
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
The element in the set {a,b}{a,b} is also in the set {b,c,a}{b,c,a}
∴{a,b}⊂{b,c,a}∴{a,b}⊂{b,c,a}
∴∴The given statement is false
ii. {a,e}⊂{x:xis an vowel in English alpahbet}{a,e}⊂{x:xis an vowel in English alpahbet}
Ans-
Given that,
{a,e}⊂{x:xis an vowel in English alpahbet}{a,e}⊂{x:xis an vowel in English alpahbet}
To examine whether the above statement is true or false
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
The element in the set {a,e}{a,e} is also in the set {a,e,i,o,u}{a,e,i,o,u}
∴{a,e}⊂{x:xis an vowel in English alpahbet}∴{a,e}⊂{x:xis an vowel in English alpahbet}
∴∴The given statement is true.
ii. {1,2,3}⊂{1,3,5}{1,2,3}⊂{1,3,5}
Ans-
Given that,
{1,2,3}⊂{1,3,5}{1,2,3}⊂{1,3,5}
To examine whether the above statement is true or false
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
The element in the set {1,2,3}{1,2,3} is not in the set {1,3,5}{1,3,5} since 2∈{1,2,3}2∈{1,2,3} and 2∉{1,3,5}2∉{1,3,5}
{1,2,3}⊄{1,3,5}{1,2,3}⊄{1,3,5}
∴∴The given statement is false.
iii. {a}⊂{a,b,c}{a}⊂{a,b,c}
Ans-
Given that,
{a}⊂{a,b,c}{a}⊂{a,b,c}
To examine whether the above statement is true or false
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
The element in the set {a}{a} is also in the set {a,b,c}{a,b,c}
∴{a}⊂{a,b,c}∴{a}⊂{a,b,c}
∴∴The given statement is true.
iv. {a}∈{a,b,c}{a}∈{a,b,c}
Ans-
Given that,
{a}∈{a,b,c}{a}∈{a,b,c}
To examine whether the above statement is true or false
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
The element in the set {a}{a} and the elements in the set {a,b,c}{a,b,c} are a,b,ca,b,c
∴{a}⊂{a,b,c}∴{a}⊂{a,b,c}
∴∴The given statement is false.
v. {x:xis an even natural less than 6}⊂{x:xis an even natural less than 6}⊂ {x:xis a natural number which divide 36}{x:xis a natural number which divide 36}
Ans-
Given that,
{x:xis an even natural less than 6}⊂{x:xis an even natural less than 6}⊂ {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
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
{x:xis an even natural less than 6}={2,4}{x:xis an even natural less than 6}={2,4}
{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} ∴{x:xis an even natural less than 6}⊂∴{x:xis an even natural less than 6}⊂ {x:xis a natural number which divide 36}{x:xis a natural number which divide 36}
∴∴The given statement is true.
3. Let A={1,2,{3,4},5}A={1,2,{3,4},5}. Which of the following statements are incorrect and why?
i. {3,4}⊂A{3,4}⊂A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if {3,4}⊂A{3,4}⊂A is correct or incorrect.
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
From the above statement,
3∈{3,4}3∈{3,4}, however 3∉A3∉A
∴∴The given statement {3,4}⊂A{3,4}⊂A is incorrect
ii. {3,4}∈A{3,4}∈A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if {3,4}∈A{3,4}∈A is correct or incorrect.
From the above statement,
{3,4}{3,4} is an element of A.
∴{3,4}∈A∴{3,4}∈A
∴∴The given statement is correct.
iii. {{3,4}}⊂A{{3,4}}⊂A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if {{3,4}}⊂A{{3,4}}⊂A is correct or incorrect.
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
From the above statement,
{3,4}∈{{3,4}}{3,4}∈{{3,4}} so that {{3,4}}∈A{{3,4}}∈A
∴{{3,4}}⊂A∴{{3,4}}⊂A
∴∴The given statement {{3,4}}⊂A{{3,4}}⊂A is correct.
iv. 1∈A1∈A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if 1∈A1∈A is correct or incorrect.
From the above statement,
11 is an element of A.
∴∴The statement 1∈A1∈A is a correct statement.
v. 1⊂A1⊂A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if 1⊂A1⊂A is correct or incorrect.
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
From the above statement,
An element of a set can never be a subset of itself. So 1⊄A1⊄A
∴∴The given statement 1⊂A1⊂A is an incorrect statement.
vi. {1,2,5}⊂A{1,2,5}⊂A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if {1,2,5}⊂A{1,2,5}⊂A is correct or incorrect.
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
From the above statement,
The each element of {1,2,5}{1,2,5} is also an element of A, So {1,2,5}⊂A{1,2,5}⊂A
∴∴The given statement {1,2,5}⊂A{1,2,5}⊂A is a correct statement
vii. {1,2,5}∈A{1,2,5}∈A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if {1,2,5}⊂A{1,2,5}⊂A is correct or incorrect.
From the above statement,
Element of {1,2,5}{1,2,5} is not an element of A, So {1,2,5}∉A{1,2,5}∉A
So the given statement {1,2,5}∈A{1,2,5}∈A is an incorrect statement.
viii. {1,2,3}⊂A{1,2,3}⊂A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if {1,2,3}⊂A{1,2,3}⊂A is correct or incorrect.
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
From the above statement, we notice that,
3∈{1,2,3}3∈{1,2,3}but 3∉A3∉A
{1,2,3}⊄A{1,2,3}⊄A
∴∴The given statement {1,2,3}⊂A{1,2,3}⊂A is an incorrect statement.
ix. ∅∈A∅∈A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if ∅∈A∅∈A is correct or incorrect.
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
From the above statement,
∅∅ is not an element of A. So, ∅∉A∅∉A
∴∴The given statement ∅∈A∅∈A is an incorrect statement.
x. ∅⊂A∅⊂A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if ∅⊂A∅⊂A is correct or incorrect.
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
From the above statement,
Since ∅∅ is a subset of every set, ∅⊂A∅⊂A
∴∴The given statement ∅⊂A∅⊂A is a correct statement.
xi. {∅}⊂A{∅}⊂A
Ans-
Given that,
A={1,2,{3,4},5}A={1,2,{3,4},5}
To find if {∅}⊂A{∅}⊂A is correct or incorrect.
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
From the above statement,
∅∅ is an element of A and it is not a subset of A.
∴∴The given statement {∅}⊂A{∅}⊂A is an incorrect statement.
3. Write down all the subsets of the following sets:
i. {a}{a}
Ans-
Given that,
{a}{a}
To write the subset of the given sets
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
Subsets of {a}{a} are ∅∅ and {a}{a}
ii. {a,b}{a,b}
Ans-
Given that,
{a,b}{a,b}
To write the subset of the given sets
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
Subsets of {a,b}{a,b} are ∅∅ and {a},{b},{a,b}{a},{b},{a,b}
iii. {1,2,3}{1,2,3}
Ans-
Given that,
{1,2,3}{1,2,3}
To write the subset of the given sets
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
Subsets of {1,2,3}{1,2,3} are ∅∅,{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}
iv. ∅∅
Ans-
Given that,
∅∅
To write the subset of the given sets
A set A is said to be a subset of B if every element of A is also an element of B
A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B
Subsets of ∅∅ is ∅∅.
4. How many elements has P(A)P(A), if A=∅A=∅?
Ans-
Given that,
A=∅A=∅
To find the number of elements does the P(A)P(A) contain
The collection of all subsets of a set A is called a power set of A and is denoted by P(A)P(A).
We know that if AA is a set with mmelements, that is, n(A)=mn(A)=m, then n[p(A)]=2mn[p(A)]=2m
If A=∅A=∅ then n(A)=0n(A)=0
n[P(A)]=20n[P(A)]=20
=1=1
∴P(A)∴P(A) has only one element.
5. Write the following as intervals
i. {x:x∈R,−4<x≤6}{x:x∈R,−4<x≤6}
Given that,
{x:x∈R,−4<x≤6}{x:x∈R,−4<x≤6}
To write the above expression as intervals
The 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]
∴{x:x∈R,−4<x≤6}=(−4,6]∴{x:x∈R,−4<x≤6}=(−4,6]
ii. {x:x∈R,−12<x<−10}{x:x∈R,−12<x<−10}
Ans-
Given that,
{x:x∈R,−12<x<−10}{x:x∈R,−12<x<−10}
To write the above expression as intervals
The 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]
∴{x:x∈R,−12<x<−10}=(−12,−10)∴{x:x∈R,−12<x<−10}=(−12,−10)
iii. {x:x∈R,0≤x<7}{x:x∈R,0≤x<7}
Ans-
Given that,
{x:x∈R,0≤x<7}{x:x∈R,0≤x<7}
To write the above expression as intervals
The 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]
∵{x:x∈R,0≤x<7}=[0,7)∵{x:x∈R,0≤x<7}=[0,7)
iv. {x:x∈R,3≤x≤4}{x:x∈R,3≤x≤4}
Ans-
Given that,
{x:x∈R,3≤x≤4}{x:x∈R,3≤x≤4}
To write the above expression as intervals
The 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]
∴{x:x∈R,3≤x≤4}=[3,4]∴{x:x∈R,3≤x≤4}=[3,4]
6. Write the following intervals in set builder form.
i. (−3,0)(−3,0)
Ans-
Given that,
(−3,0)(−3,0)
To write the above interval in set builder form
The 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]
∴(−3,0)={x:x∈R,−3<x<0}∴(−3,0)={x:x∈R,−3<x<0}
ii. [6,12][6,12]
Ans-
Given that,
[6,12][6,12]
To write the above interval in set builder form
The 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]
∴[6,12]={x:x∈R,6≤x≤12}∴[6,12]={x:x∈R,6≤x≤12}
iii. (6,12](6,12]
Ans-
Given that,
(6,12](6,12]
To write the above interval in set builder form
The 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]
∴(6,12]={x:x∈R,6<x≤12}∴(6,12]={x:x∈R,6<x≤12}
iv. [−23,5)[−23,5)
Ans-
Given that,
[−23,5)[−23,5)
To write the above interval in set builder form
The 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]
∴[−23,5)={x:x∈R,−23≤x<5}∴[−23,5)={x:x∈R,−23≤x<5}
7. What universal set(s) would you propose for each of the following:
i. The set of right triangles
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.
ii. The set of isosceles triangles
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.
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?
i. {0,1,2,3,4,5,6}{0,1,2,3,4,5,6}
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⊂A⊂ {0,1,2,3,4,5,6}{0,1,2,3,4,5,6} B⊂B⊂ {0,1,2,3,4,5,6}{0,1,2,3,4,5,6} C⊄C⊄ {0,1,2,3,4,5,6}{0,1,2,3,4,5,6} ∴∴The 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
ii. ∅∅
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 ∅∅ is the universal set of A, B and C It can be observed that, A⊄∅A⊄∅ B⊄∅B⊄∅ C⊄∅C⊄∅ ∴∴The set ∅∅ cannot be an universal set for A, B and C.
iii. {0,1,2,3,4,5,6,7,8,9,10}{0,1,2,3,4,5,6,7,8,9,10}
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⊂A⊂ {0,1,2,3,4,5,6,7,8,9,10}{0,1,2,3,4,5,6,7,8,9,10} B⊂B⊂ {0,1,2,3,4,5,6,7,8,9,10}{0,1,2,3,4,5,6,7,8,9,10} C⊂C⊂ {0,1,2,3,4,5,6,7,8,9,10}{0,1,2,3,4,5,6,7,8,9,10} ∴∴The 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
iv. {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8}
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⊂A⊂ {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8} B⊂B⊂ {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8} C⊄C⊄ {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8} ∴∴The 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
Exercise (1.4)
1. Find the union of each of following pair of sets
i. X={1,3,5},Y={1,2,3}X={1,3,5},Y={1,2,3}
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∪Y={1,3,5}∪{1,2,3}X∪Y={1,3,5}∪{1,2,3} ∴X∪Y={1,2,3,5}∴X∪Y={1,2,3,5}
ii. A={a,e,i,o,u},B={a,b,c}A={a,e,i,o,u},B={a,b,c}
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∪B={a,e,i,o,u}∪{a,b,c}A∪B={a,e,i,o,u}∪{a,b,c} ∴A∪B={a,b,c,e,i,o,u}∴A∪B={a,b,c,e,i,o,u}
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}
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∪B={3,6,9,…}∪{1,2,3,4,5,6}A∪B={3,6,9,…}∪{1,2,3,4,5,6} ={1,2,3,4,5,6,9,12,15…}={1,2,3,4,5,6,9,12,15…} ∴A∪B∴A∪B ={1,2,3,4,5,6,9,12,15…}={1,2,3,4,5,6,9,12,15…}
iv. A={x:xis a natural number 1x≤6},A={x:xis a natural number 1x≤6}, B={x:xis a natural number 6×10}B={x:xis a natural number 6×10}
Ans- Given that, A={x:xis a natural number 1x≤6},A={x:xis a natural number 1x≤6}, 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≤6}={2,3,4,5,6}A={x:xis a natural number 1x≤6}={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∪B={2,3,4,5,6}∪{7,8,9}A∪B={2,3,4,5,6}∪{7,8,9} ∴A∪B={2,3,4,5,6,7,8,9}∴A∪B={2,3,4,5,6,7,8,9}
v. A={1,2,3},B=∅A={1,2,3},B=∅
Ans- Given that, A={1,2,3},B=∅A={1,2,3},B=∅ 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∪B={1,2,3}∪∅A∪B={1,2,3}∪∅ ∴A∪B={1,2,3}∴A∪B={1,2,3}
2. Let A={a,b}A={a,b} and B={a,b,c}B={a,b,c}. Is A⊂BA⊂B? What is A∪BA∪B?
Ans- Given that, A={a,b}A={a,b} and B={a,b,c}B={a,b,c} To find if A⊂BA⊂B and A∪BA∪B A set A is said to be a subset of B if every element of A is also an element of B A⊂BA⊂B if a∈A,a∈Ba∈A,a∈B It can be observed that A⊂BA⊂B A∪B={a,b}∪{a,b,c}A∪B={a,b}∪{a,b,c} ∴A∪B={a,b,c}∴A∪B={a,b,c}
3. If A and B are two sets such that A⊂BA⊂B then what is A⋃BA⋃B
Ans- Given that, A and B are two sets To find A∪BA∪B when A⊂BA⊂B If A and B are two sets such that A⊂BA⊂B, then A∪B=BA∪B=B
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
i. A∪BA∪B
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∪BA∪B 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∪B={1,2,3,4}∪{3,4,5,6}A∪B={1,2,3,4}∪{3,4,5,6} ∴A∪B={1,2,3,4,5,6}∴A∪B={1,2,3,4,5,6}
ii. A∪CA∪C
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∪CA∪C 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∪C={1,2,3,4}∪{5,6,7,8}A∪C={1,2,3,4}∪{5,6,7,8} ∴A∪C={1,2,3,4,5,6,7,8}∴A∪C={1,2,3,4,5,6,7,8}
iii. B∪CB∪C
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∪CB∪C 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∪C={3,4,5,6}∪{5,6,7,8}B∪C={3,4,5,6}∪{5,6,7,8} ∴B∪C={3,4,5,6,7,8}∴B∪C={3,4,5,6,7,8}
iii. B∪DB∪D
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∪DB∪D 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∪D={3,4,5,6}∪{7,8,9,10}B∪D={3,4,5,6}∪{7,8,9,10} ∴B∪D={3,4,5,6,7,8,9,10}∴B∪D={3,4,5,6,7,8,9,10}
iv. A∪B∪CA∪B∪C
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∪B∪CA∪B∪C 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∪B∪C={1,2,3,4}∪{3,4,5,6}∪{5,6,7,8}A∪B∪C={1,2,3,4}∪{3,4,5,6}∪{5,6,7,8} ∴A∪B∪C={1,2,3,4,5,6,7,8}∴A∪B∪C={1,2,3,4,5,6,7,8}
v. A∪B∪DA∪B∪D
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∪B∪DA∪B∪D 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∪B∪D={1,2,3,4}∪{3,4,5,6}∪{7,8,9,10}A∪B∪D={1,2,3,4}∪{3,4,5,6}∪{7,8,9,10} ∴A∪B∪D={1,2,3,4,5,6,7,8,9,10}∴A∪B∪D={1,2,3,4,5,6,7,8,9,10}
vi. B∪C∪DB∪C∪D
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∪C∪DB∪C∪D 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∪C∪D={3,4,5,6}∪{5,6,7,8}∪{7,8,9,10}B∪C∪D={3,4,5,6}∪{5,6,7,8}∪{7,8,9,10} ∴B∪C∪D={3,4,5,6,7,8,9,10}∴B∪C∪D={3,4,5,6,7,8,9,10}
5. Find the intersection of each pair of sets:
i. X={1,3,5},Y={1,2,3}X={1,3,5},Y={1,2,3}
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∩Y={1,3,5}∪{1,2,3}X∩Y={1,3,5}∪{1,2,3} ∴X∩Y={1,3}∴X∩Y={1,3}
ii. A={a,e,i,o,u},B={a,b,c}A={a,e,i,o,u},B={a,b,c}
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∩B={a,e,i,o,u}∪{a,b,c}A∩B={a,e,i,o,u}∪{a,b,c} ∴A∩B={a}∴A∩B={a}
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}
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∩B={3,6,9,…}∩{1,2,3,4,5,6}A∩B={3,6,9,…}∩{1,2,3,4,5,6} ={3}={3} ∴A∩B∴A∩B ={3}={3}
iv. A={x:xis a natural number 1x≤6},A={x:xis a natural number 1x≤6}, B={x:xis a natural number 6×10}B={x:xis a natural number 6×10}
Ans- Given that, A={x:xis a natural number 1x≤6},A={x:xis a natural number 1x≤6}, 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≤6}={2,3,4,5,6}A={x:xis a natural number 1x≤6}={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∩B={2,3,4,5,6}∩{7,8,9}A∩B={2,3,4,5,6}∩{7,8,9} ∴A∩B=∅∴A∩B=∅
v. A={1,2,3},B=∅A={1,2,3},B=∅
Ans- Given that, A={1,2,3},B=∅A={1,2,3},B=∅ 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⋂B={1,2,3}∩∅A⋂B={1,2,3}∩∅ ∴A∩B=∅∴A∩B=∅
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
i. A∩BA∩B
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∩BA∩B The intersection of sets A and B is the set of all elements which are common to both A and B. A∩B={3,5,7,9,11}∩{7,9,11,13}A∩B={3,5,7,9,11}∩{7,9,11,13} ∴A∩B={7,9,11}∴A∩B={7,9,11}
ii. B∩CB∩C
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∩CB∩C The intersection of sets A and B is the set of all elements which are common to both A and B. B∩C={7,9,11,13}∩{11,13,15}B∩C={7,9,11,13}∩{11,13,15} ∴B∩C={11,13}∴B∩C={11,13}
iii. A∩C∩DA∩C∩D
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∩C∩DA∩C∩D The intersection of sets A and B is the set of all elements which are common to both A and B. A∩C∩D={3,5,7,9,11}∩{11,13,15}∩{15,17}A∩C∩D={3,5,7,9,11}∩{11,13,15}∩{15,17} ∴A∩C∩D=∅∴A∩C∩D=∅
iv. A∩CA∩C
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∩CA∩C The intersection of sets A and B is the set of all elements which are common to both A and B. A∩C={3,5,7,9,11}∩{11,13,15}A∩C={3,5,7,9,11}∩{11,13,15} ∴A∩C={11}∴A∩C={11}
v. B∩DB∩D
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∩DB∩D The intersection of sets A and B is the set of all elements which are common to both A and B. B∩D={7,9,11,13}∩{15,17}B∩D={7,9,11,13}∩{15,17} ∴B∩D=∅∴B∩D=∅
vi. A∩(B⋃C)A∩(B⋃C)
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∩(B∪C)A∩(B∪C) The intersection of sets A and B is the set of all elements which are common to both A and B. A∩(B∪C)=(A∩B)∪(A∩C)A∩(B∪C)=(A∩B)∪(A∩C) A∩B={3,5,7,9,11}∩{7,9,11,13}A∩B={3,5,7,9,11}∩{7,9,11,13} A∩B={7,9,11}A∩B={7,9,11} A∩D={11}A∩D={11} A∩(B∪C)={7,9,11}∪{11}A∩(B∪C)={7,9,11}∪{11} ={11}={11} ∴A∩(B∪C)={11}∴A∩(B∪C)={11}
vii. A∩DA∩D
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∩DA∩D The intersection of sets A and B is the set of all elements which are common to both A and B. A∩D={3,5,7,9,11}∩{15,17}A∩D={3,5,7,9,11}∩{15,17} ∴A∩D=∅∴A∩D=∅
viii. A∩(B∪D)A∩(B∪D)
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∩(B∪D)A∩(B∪D) The intersection of sets A and B is the set of all elements which are common to both A and B. A∩(B∪D)=(A∩B)∪(A∩D)A∩(B∪D)=(A∩B)∪(A∩D) A∩B={3,5,7,9,11}∩{7,9,11,13}A∩B={3,5,7,9,11}∩{7,9,11,13} A∩D=∅A∩D=∅ ∴A∩(B∪D)={7,9,11}∪∅∴A∩(B∪D)={7,9,11}∪∅ ={7,9,11}={7,9,11}
ix. (A∩B)∩(B∪C)(A∩B)∩(B∪C)
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∩B)∩(B∪C)(A∩B)∩(B∪C) The intersection of sets A and B is the set of all elements which are common to both A and B. A∩B={3,5,7,9,11}∩{7,9,11,13}A∩B={3,5,7,9,11}∩{7,9,11,13} A∩B={7,9,11}A∩B={7,9,11} B∪C={7,9,11,13}∪{11,13,15}B∪C={7,9,11,13}∪{11,13,15} ={7,9,11,13,15}={7,9,11,13,15} (A∩B)∩(B∪C)={7,9,11}∩{7,9,11,13,15}(A∩B)∩(B∪C)={7,9,11}∩{7,9,11,13,15} ={7,9,11}={7,9,11}
x. (A∪D)∩(B∪C)(A∪D)∩(B∪C)
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∪D)∩(B∪C)(A∪D)∩(B∪C) The intersection of sets A and B is the set of all elements which are common to both A and B. A∩D={3,5,7,9,11}∩{15,17}A∩D={3,5,7,9,11}∩{15,17} A∩D={3,5,7,9,11,15,17}A∩D={3,5,7,9,11,15,17} B∪C={7,9,11,13}∪{11,13,15}B∪C={7,9,11,13}∪{11,13,15} ={7,9,11,13,15}={7,9,11,13,15} (A∪D)∩(B∪C)={3,5,7,9,11,15,17}∩{7,9,11,13,15}(A∪D)∩(B∪C)={3,5,7,9,11,15,17}∩{7,9,11,13,15} ∴(A∪D)∩(B∪C)={7,9,11,15}∴(A∪D)∩(B∪C)={7,9,11,15}
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
i. A∩BA∩B
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∩BA∩B The intersection of sets A and B is the set of all elements which are common to both A and B. A∩B={1,2,3,4,…}∩{2,4,6,8…}A∩B={1,2,3,4,…}∩{2,4,6,8…} ∴A∩B=B∴A∩B=B ={x:xis an even natural number}={x:xis an even natural number}
ii. A∩CA∩C
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∩CA∩C The intersection of sets A and B is the set of all elements which are common to both A and B. A∩C={1,2,3,4,…}∩{1,3,5,7…}A∩C={1,2,3,4,…}∩{1,3,5,7…} ∴A∩C=C∴A∩C=C {x:xis an odd natural number}{x:xis an odd natural number}
iii. A∩DA∩D
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∩DA∩D The intersection of sets A and B is the set of all elements which are common to both A and B. A∩D={1,2,3,4,…}∩{2,3,5,7,…}A∩D={1,2,3,4,…}∩{2,3,5,7,…} ∴A∩D=D∴A∩D=D {x:xis a prime number}{x:xis a prime number}
iv. B∩CB∩C
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∩CB∩C The intersection of sets A and B is the set of all elements which are common to both A and B. B∩C={2,4,6,8…}∩{1,3,5,7…}B∩C={2,4,6,8…}∩{1,3,5,7…} ∴B∩C=∅∴B∩C=∅
v. B∩DB∩D
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∩DB∩D The intersection of sets A and B is the set of all elements which are common to both A and B. B∩D={2,4,6,8,…}∩{2,3,5,7,…}B∩D={2,4,6,8,…}∩{2,3,5,7,…} ∴A∩D={2}∴A∩D={2}
vi. C∩DC∩D
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∩DC∩D The intersection of sets A and B is the set of all elements which are common to both A and B. C∩D={1,3,5,7,…}∩{2,3,5,7,…}C∩D={1,3,5,7,…}∩{2,3,5,7,…} ∴C∩D={x:xis a odd prime number}∴C∩D={x:xis a odd prime number}
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