Question

if n(a)=10 and n(anb)=3,find n((anb)'na)

Answer 1

Answer:

30

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Answer 2

The correct answer  is n((A' ∪ B')∩A) = 7.

Given:

Two sets A and B such that

n(A) = 10 and n(A∩B) = 3

To Find:

n((A∩B)' ∩ A).

Solution:

To solve this problem we will use the following concepts:

i) A∪Ф = A i.e. the union of a set A with an empty set will give us the set A.

ii) A∩Ф = Ф i.e. the intersection of a set A with an empty set will give us an empty set

iii) From De Morgan's law, we have

(A∩B)' = A' ∪ B'

i.e. the complement of the intersection of two sets is the union of the complement of the two sets.

iv) (B'∩A) = A - (A∩B)

We need to find n((A'∪B') ∩ A).

We have been given two sets A and B such that

n(A) = 10 and n(A∩B) = 3.

Now, ((A∩B)' ∩ A) = (A' ∪ B')∩A            ............. from De Morgan's law in (iii)

On taking A inside the brackets, we have

((A∩B)' ∩ A) = (A'∩A) ∪ (B'∩A) = Ф ∪ (B'∩A) = (B'∩A)   .............from (ii) and (iii)

⇒ ((A∩B)' ∩ A) = A - (A∩B)                                           ................... from (iv)

⇒ n((A' ∪ B') ∩ A) = n(A) - n(A∩B)

⇒  n((A' ∪ B') ∩ A) = 10 - 3 = 7

∴ n((A' ∪ B') ∩ A) = 7.

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