Question

For any sets A and B, prove that

$(A\times B)\cap (B\times A)=(A\cap B)\times (B\cap A)$.

Answer 1

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

$(a,b)ϵ(A\times B)\cap (B\times A)\Rightarrow (a,b)ϵA\times B$ and $(a,b)ϵB\times A$

$\Rightarrow \left(aϵA\text{and}bϵB\right)$ and $\Rightarrow \left(aϵB\text{and}bϵA\right)$

$\Rightarrow \left(aϵA\text{and}aϵB\right)$ and $\Rightarrow \left(bϵB\text{and}bϵA\right)$

$\Rightarrow aϵ(A\cap B)$ and $bϵ(B\cap A)$

$\Rightarrow (a,b)ϵ(A\cap B)\times (B\cap A)$

$\therefore (A\times B)\cap (B\times A)\subseteq (A\cap B)\times (B\cap A)\dots \dots \left(i\right)$

Again, let (x, y) be an arbitrary element of $(A\cap B)\times (B\cap A)$. Then,

$(x,y)ϵ(A\cap B)\times (B\cap A)\Rightarrow xϵA\cap B$ and $yϵB\cap A$

$\Rightarrow (x,y)ϵ(A\cap B)\times (B\cap A)\Rightarrow xϵA\cap B$ and $yϵB\cap A$

$\Rightarrow \left(xϵA\text{and}xϵB\right)$ and $yϵB\text{and}yϵA$

$\Rightarrow \left(xϵA\text{and}yϵB\right)$ and $xϵB\text{and}yϵA$

$\Rightarrow (x,y)ϵA\times B$ and $(x,y)ϵB\times A$

$\Rightarrow (x,y)ϵ(A\times B)\cap (B\times A)$

$\therefore (A\cap B)\times (B\cap A)\subseteq (A\times B)\cap (B\times A)\dots \dots \left(ii\right)$

Thus, from (i) and (ii), we get

$(A\times B)\cap (B\times A)=(A\cap B)\times (B\times A)$.