Question

if a and b be two non-void sets with n elements in common then prove that a x b & b x a will have n2 elements in common.

Answer 1

(AxB) ∩ (CxD) = (A∩C) x (B∩D)

So for this we will get

(AxB) ∩ (BxA) = (A∩B) x (B∩A)

n(A∩B) = n elements

n [(A∩B) x (B∩A)] = n (A∩B) . n(B∩A) = n.n = n² elements

by (i), n [(AxB) ∩ (BxA)] = n² elements

So for this A x B and Bx A have n² elements in common