Question

Comment on the following statements
$A\times (B\cap C)=(A\times B)\cap (A\times C)$

Answer 1

$(x,y)\in A\times (B\cap C)$
$\iff (x\in A)$ and $(y\in B\cap C)$ (by def. of cross product)
$\iff (x\in A)$ and $(y\in Bandy\in C)$
$\iff (x\in Aandy\in B)$ and $(x\in Aandy\in C)$
$\iff (x,y)\in (A\times B)$ and $(x,y)\in (A\times C)$
$\iff (x,y)\in (A\times B)\cap (A\times C)$
Hence $A\times (B\cap C)=(A\times B)\cap (A\times C)$