Answer to Question #299944 in Discrete Mathematics for hudayfi

Question

Let A and B be two non-empty sets. Show that A × B = B × A if and only if A = B.

Answer 1

Solution:

A=ϕ,B=ϕ[given]

 Let

 a∈A,b∈B..............(i)

 i)a∈A(fix)

 (a,b)∈A×B

 (a,b)∈B×A

 ⇒a∈B,b∈A∀b∈B

 ⇒B⊆A.........(ii)

 ii)b∈B(fix)

 (a,b)∈A×B

 (a,b)∈B×A

 ⇒b∈A,a∈B∀a∈A

 ⇒A⊆B.........(ii)

 from(i)and(ii)

 A=B