If A = {4, 5, 6, 7, 8} and B = {6, 8, 10, 12}, find :
(i) A ∪ B
(ii) A ∩ B
(iii) A − B
(iv) B − A.
Answer
A = {4, 5, 6, 7, 8} and B = {6, 8, 10, 12}.
(i) A ∪ B = {all elements of A and of B} = {4, 5, 6, 7, 8, 10, 12}
(ii) A ∩ B = {elements common to A and B} = {6, 8}
(iii) A − B = {elements of A which are not in B} = {4, 5, 7}
(iv) B − A = {elements of B which are not in A} = {10, 12}
If A = {3, 5, 7, 9, 11} and B = {4, 7, 10}, find :
(i) n(A)
(ii) n(B)
(iii) A ∪ B and n(A ∪ B)
(iv) A ∩ B and n(A ∩ B)
Answer
A = {3, 5, 7, 9, 11} and B = {4, 7, 10}.
(i) A has 5 elements. n(A) = 5
(ii) B has 3 elements. n(B) = 3
(iii) A ∪ B = {3, 4, 5, 7, 9, 10, 11}
A ∪ B = {3, 4, 5, 7, 9, 10, 11} and n(A ∪ B) = 7
(iv) A ∩ B = {elements common to A and B} = {7}
A ∩ B = {7} and n(A ∩ B) = 1
If A = {2, 4, 6, 8} and B = {3, 6, 9, 12}, find :
(i) (A ∩ B) and n(A ∩ B)
(ii) (A − B) and n(A − B)
(iii) n(B)
Answer
A = {2, 4, 6, 8} and B = {3, 6, 9, 12}.
(i) A ∩ B = {elements common to A and B} = {6}
A ∩ B = {6} and n(A ∩ B) = 1
(ii) A − B = {elements of A which are not in B} = {2, 4, 8}
A − B = {2, 4, 8} and n(A − B) = 3
(iii) B has 4 elements. n(B) = 4
If P = {x : x is a factor of 12} and Q = {x : x is a factor of 16}, find :
(i) n(P)
(ii) n(Q)
(iii) Q − P and n(Q − P)
Answer
P = {x : x is a factor of 12} = {1, 2, 3, 4, 6, 12}.
Q = {x : x is a factor of 16} = {1, 2, 4, 8, 16}.
(i) P has 6 elements. n(P) = 6
(ii) Q has 5 elements. n(Q) = 5
(iii) Q − P = {elements of Q which are not in P} = {8, 16}
Q − P = {8, 16} and n(Q − P) = 2
M = {x : x is a natural number between 0 and 8} and N = {x : x is a natural number from 5 to 10}. Find :
(i) M − N and n(M − N)
(ii) N − M and n(N − M)
Answer
M = {x : x is a natural number between 0 and 8} = {1, 2, 3, 4, 5, 6, 7}.
N = {x : x is a natural number from 5 to 10} = {5, 6, 7, 8, 9, 10}.
(i) M − N = {elements of M which are not in N} = {1, 2, 3, 4}
M − N = {1, 2, 3, 4} and n(M − N) = 4
(ii) N − M = {elements of N which are not in M} = {8, 9, 10}
N − M = {8, 9, 10} and n(N − M) = 3
If A = {x : x is a natural number divisible by 2 and x < 16} and B = {x : x is a whole number divisible by 3 and x < 18}, find :
(i) n(A)
(ii) n(B)
(iii) A ∩ B and n(A ∩ B)
(iv) n(A − B)
Answer
A = {x : x is a natural number divisible by 2 and x < 16} = {2, 4, 6, 8, 10, 12, 14}.
B = {x : x is a whole number divisible by 3 and x < 18} = {0, 3, 6, 9, 12, 15}.
(i) A has 7 elements. n(A) = 7
(ii) B has 6 elements. n(B) = 6
(iii) A ∩ B = {elements common to A and B} = {6, 12}
A ∩ B = {6, 12} and n(A ∩ B) = 2
(iv) A − B = {elements of A which are not in B} = {2, 4, 8, 10, 14}
n(A − B) = 5
Let A and B be two sets such that n(A) = 75, n(B) = 65 and n(A ∩ B) = 45, find :
(i) n(A ∪ B)
(ii) n(A − B)
(iii) n(B − A)
Answer
Given n(A) = 75, n(B) = 65 and n(A ∩ B) = 45.
(i) n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
= 75 + 65 − 45
= 140 − 45
n(A ∪ B) = 95
(ii) n(A − B) = n(A) − n(A ∩ B)
= 75 − 45
n(A − B) = 30
(iii) n(B − A) = n(B) − n(A ∩ B)
= 65 − 45
n(B − A) = 20
Let A and B be two sets such that n(A) = 45, n(B) = 38 and n(A ∪ B) = 70, find :
(i) n(A ∩ B)
(ii) n(A − B)
(iii) n(B − A)
Answer
Given n(A) = 45, n(B) = 38 and n(A ∪ B) = 70.
(i) n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
= 45 + 38 − 70
= 83 − 70
n(A ∩ B) = 13
(ii) n(A − B) = n(A) − n(A ∩ B)
= 45 − 13
n(A − B) = 32
(iii) n(B − A) = n(B) − n(A ∩ B)
= 38 − 13
n(B − A) = 25
Let n(A) = 30, n(B) = 27 and n(A ∪ B) = 45, find :
(i) n(A ∩ B)
(ii) n(A − B)
Answer
Given n(A) = 30, n(B) = 27 and n(A ∪ B) = 45.
(i) n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
= 30 + 27 − 45
= 57 − 45
n(A ∩ B) = 12
(ii) n(A − B) = n(A) − n(A ∩ B)
= 30 − 12
n(A − B) = 18
Let n(A) = 31, n(B) = 20 and n(A ∩ B) = 6, find :
(i) n(A − B)
(ii) n(B − A)
(iii) n(A ∪ B)
Answer
Given n(A) = 31, n(B) = 20 and n(A ∩ B) = 6.
(i) n(A − B) = n(A) − n(A ∩ B)
= 31 − 6
n(A − B) = 25
(ii) n(B − A) = n(B) − n(A ∩ B)
= 20 − 6
n(B − A) = 14
(iii) n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
= 31 + 20 − 6
= 51 − 6
n(A ∪ B) = 45