(a) n(A) and n(Ф)

    (b) n(A U Ф) and n(A n Ф)

Solution: Here, A = {x : x ∈ N and x ≤ 10 } = {1,2,3,….,10}

and Ф is the empty set (means no any elements).

.’. (a) n(A) = 10  and n(Ф) = 0

(b) n(A ∪ Ф) = n(A) = 10

and n(A n Ф) = n(Ф) = 0

 (i) n(A ∪ B), n(A – B) and n(A’ n B’)

 (ii) n(B’), n(A’), n(A ∪ B)’

 (iii) n(P ∪ Q) and n(P n Q) where P= A – B and Q = A n B

 (iv) how many elements are there in the set U – (A ∪ B)

Solutions: Given, n(U) = 120 , n(A) = 42 , n(B) = 50 and n(A n B) = 21

(i) n(A ∪ B) = n(A) + n(B) – n(A n B)

= 42 + 50 – 21 = 71

n(A – B) = n(A) – n(A n B) = 42 – 21 = 21

n(B – A) = n(B) – n(A n B) = 50 -21 = 29

(ii) n(B’) = n(U) – n(B) = 120 – 50 = 70

n(A’) = n(U) – n(A) = 120 – 42 = 78

  n(A ∪ B)’ = n(U) – n(A U B) = 120 – 71 = 49

(iii) n(P U Q) = n{(A – B) U (A n B)}

=n{(A n B’) U (A n B)} = n{A U (B’ n B)}

=n(A U Ф) = n(A) = 42

And, n(P n Q) = n{(A – B) n (A n B)}

= n{(A n B’) n (A n B)} = n{A n (B’ n B)}

= n(A n Ф) = n(Ф) = 0

(iii) n{U – (A U B)} = n(U) – n{U n (A U B)}

=n(U) – n(A U B) = 120 – 71 = 49

Solutions: Given, n(A n B) = 36, n(A – B) = 25, n(B – A) = 20

.’. n(A U B) = n(A – B) + n(B – A) + n(A n B)

= 25 + 20 + 36 = 81`

Secondly, n(A) = n(A – B) + n(A n B) = 25 + 36 = 61

Lastly, n(B) = (B – A) + n(A n B) = 20 + 36 = 56