Question

Show that it is impossible to have sets A and B such that set A has 32 elements set B has 42 elements, A $\cap$ B has 12 elements and A $\cup$ B has 64 elements.

Answer 1

Let n(A) = 32, n(B) = 42 and n(A$\cap$B) = 12

We know that n(A$\cup$B) = n(A) + n(B) $-$ n(A$\cap$B)
= 32 + 42 $-$12
= 62

∴ It is not possible that A$\cup$B has 64 elements when A, B and A$\cap$B have 32, 42 and 12 elements, respectively. However, it can be possible if A$\cup$B has 62 elements.