Question

If $A=\{a,b,p,d\}$, $B=\{p,d,e\}$ , $C=\{p,e,f,g\}$ then verify:
(i) $A\times (B\cap C)=(A\times B)\cap (A\times C)$ (ii) $A\times (B-C)=(A\times B)-(A\times C)$

Answer 1

$A=\{a,b,p,d\},B=\{p,d,e\},C=\{p,e,f,g\}$
(i) $A\times B=\left\{\right(a,p),(a,d),(a,e),(b,p),(b,d),(b,e),(p,p),(p,d),(p,e),(d,p),(d,d),(d,e\left)\right\}$
$A\times C=\left\{\right(a,p),(a,e),(a,f),(a,g),(b,p),(b,e),(b,f),(b,g),(p,p),(p,e),(p,f),(p,g),(d,p),(d,e),(d,f),(d,g)\}$
$B\cap C=\{p,e\}$
$A\times (B\cap C)=\left\{\right(a,p),(a,e),(b,p),(b,e),(p,p),(p,e),(d,p),(d,e\left)\right\}$
$(A\times B)\cap (A\times C)=\left\{\right(a,p),(a,e),(b,p),(b,e),(p,p),(p,e),(d,p),(d,e\left)\right\}$
$\therefore A\times (B\cap C)=(A\times B)\cap (A\times C)$

(ii) $B-C=\left\{d\right\}$
$A\times (B-C)=\left\{\right(a,d),(b,d),(p,d),(d,d\left)\right\}$
$(A\times B)-(A\times C)=\left\{\right(a,d),(b,d),(p,d),(d,d\left)\right\}$