Question

Prove that-
(AxB) = (BxA) iff A=0(phi) B=0(phi) or A=B

Answer 1

phiphi
a:b phi(a) phi(b)

Answer 2

Step-by-step explanation:

If A = B then substituting B for A gives A × B = A × A = B × A. If A = ∅ or B = ∅,

Then by Proposition " If A is a set, then A × ∅ = ∅ and ∅ × A = ∅

We get

A × B = ∅ = B × A.

Suppose that A and B are non-empty sets and A × B = B × A. Let x ∈ A. Since

B = ∅, there exists an element y ∈ B, so that (x, y) ∈ A × B. Since A × B = B × A, we

have that ( x , y ) ∈ B × A . By the definition of Cartesian product, x ∈ B. Therefore,

A ⊆ B. Similarly B ⊆ A. Thus, A = B.