Question

If A and B are 2 sets such that A $\cup$ B has 40 elements, A has 18 elements and B has 29 elements, how many elements does A $\cap$ B have?

___

Answer 1

We know

n(A $\cup$ B) = n(A) + n(B) - n(A$\cap$B)

We already learnt that

n(A $\cup$ B) = n(A - B) + n(B - A) + n(A $\cap$ B) → because A - B, B - A and A $\cap$ B form disjoint sets, as shown in Venn Diagram,

Areas of A - B, A $\cap$ B and B - A when combined gives area of A $\cup$ B.

So, n(A $\cup$ B) = n(A - B) + n(A $\cap$ B) + n(B - A)

= n(A - B) + n(A $\cap$ B) + n(B - A) + [n(A$\cap$ B) - n(A$\cap$ B)]

= [n(A -B) + n(A$\cap$B)] + [n(B -A) + n(A$\cap$B)] - n(A$\cap$B)

= n(A) + n(B) - n(A $\cap$B)

Hence n(A $\cup$ B) = n(A) + n(B) - n(A $\cap$B)

Given n(A $\cup$ B) = 40

n(A) = 18

n(B) = 29

Hence, n(A $\cap$B) = n(A) + n(B) - n(A $\cup$ B)

= 18 + 29 - 40

= 7