Question

If $A=\{1,2,3\},B=\left\{4\right\},\left\{5\right\}$,

then verfy that:

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

Answer 1

Given : $A=\{1,2,3\},B=\left\{4\right\},\left\{5\right\}$

Now, $B\cap C=\left\{4\right\}\cap \left\{5\right\}$

$\Rightarrow B\cap C=\varphi$

$A\times (B\cap C)=\{1,2,3\}\times \varphi$

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

Also, =$\{1,2,3\}\times \left\{4\right\}$

$\Rightarrow A\times B\left\{\right(1,4),(2,4),(3,4\left)\right\}$

$\Rightarrow A\times C=\left\{\right(1,5),(2,5),(3,5\left)\right\}$

$(A\times B)\cap (A\times C)=\left\{\right(1,4),(2,4),(3,4\left)\right\}$

$\cap \left\{\right(1,5),(2,5),(3,5\left)\right\}$

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

Hence verfied.