Question

Let A and B be two sets. Show that the sets $A\times B$ and $B\times A$ have an element in common if the sets A and B have an element in common.

Answer 1

Let (a, b) be an arbitary element of $(A\times B)\cap (B\times A)$. Then,

$(a,b)ϵ(B\times A).$ Then,

$(a,b)ϵ(A\times B)\cap (B\times A)$

$\iff (a,b)ϵA\times B$ and $(a,b)ϵB\times A$

$\iff (a,ϵAandbϵB)and\left(aϵBandbϵA\right)$

$\iff (a,ϵAandaϵB)and\left(bϵAandbϵB\right)$

$\iff aϵA\cap BandbϵA\cap B$

Hence, the sets $A\times B$ and $B\times A$ have an element in common iff the sets A and have an element in common.