Question

Verify$A=\{1,2\}$
$B=\{1,2,3,4\}$$C=\{5,6\}$$D=\{5,6,7,8\}$. Prove $A\times (B\cap C)=(A\times B)\cap (A\times C)$

Answer 1

given

$A=\{1,2\},B=\{1,2,3\},C=\{5,6\},D=\{5,6,7,8\}$

therefore $(A\times B)=(1,1),(1,2)(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)$ and

$(A\times C)=(1,5),(1,6),(2,5),(2,6)$

$\Rightarrow (A\times B)\cap (A\times C)$ is a null set

$B=\{1,2,3,4\},C=\{5,6\}$

so $(B\cap C)$ is also a null set

therefore $A\times (B\cap C)$ is a null set

$\Rightarrow A\times (B\cap C)=(A\times B)\cap (A\times C)$