Question

Let $A=$ The set of all natural numbers less than $8,B=$ The set of all prime numbers less than $8,C=$ The set of even prime number. Verify that : $(A\cap B)\times C=(A\times C)\cap (B\times C)$

Answer 1

$A=\{1,2,3,4,5,6,7\}$
$B=\{2,3,5,7\}$
$C=\left\{2\right\}$
$(A\cap B)\times C=(A\times C)\cap (B\times C)$
$A\cap B=\{1,2,3,4,5,6,7\}\cap \{2,3,5,7\}$
$=\{2,3,5,7\}$
$(A\cap B)\times C=\{2,3,5,7\}\times \left\{2\right\}$
$=\left\{\right(2,2\left)\right(3,2\left)\right(5,2\left)\right(7,2\left)\right\}\dots .\left(1\right)$
$A\times C=\{1,2,3,4,5,6,7\}\times \left\{2\right\}$
$=\left\{\right(1,2\left)\right(2,2\left)\right(3,2\left)\right(4,2\left)\right(5,2\left)\right(6,2\left)\right(7,2\left)\right\}$
$B\times C=\{2,3,5,7\}\times \left\{2\right\}$
$=\left\{\right(2,2\left)\right(3,2\left)\right(5,2\left)\right(7,2\left)\right\}$
$(A\times C)\cap (B\times C)=\left\{\right(2,2\left)\right(3,2\left)\right(5,2\left)\right(7,2\left)\right\}\dots .\left(2\right)$
From (1) and $\left(2\right)$ we get
$(A\cap B)\times C=(A\times C)\cap (B\times C)$