Question

Let $A=\{1,2\},B=\{1,2,3,4\},C=\{5,6\}$ and $D=\{5,6,7,8\}$ Verify that
(i) $A\times (B\cap C)=(A\times B)\cap (A\times C)$
(ii) $A\times C$ is a subset of $B\times D$

Answer 1

(i) To verify : $A\times (B\cap C)=(A\times B)\cap (A\times C)$
We have $B\cap C=\{1,2,3,4\}\cap \{5,6\}=\varphi$
$\therefore$ L.H.S = $A\times (B\cap C)=A\times \varphi =\varphi$

$A\times B=\left\{\right(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4\left)\right\}$
$A\times C=\left\{\right(1,5),(1,6),(2,5),(2,6\left)\right\}$
$\therefore R.H.S.=(A\times B)\cap (A\times C)=\varphi$
$\therefore L.H.S=R.H.S$
Hence $A\times (B\cap C)=(A\times B)\cap (A\times C)$

(ii) To verify: $A\times C$ is a subset of $B\times D$
$A\times C=\left\{\right(1,5),(1,6),(2,5),(2,6\left)\right\}$
$B\times D=\left\{\right(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),$

$(3,8),(4,5),(4,6),(4,7),(4,8)\}$
We can observe that all the elements of set $A\times C$ are the elements of set $B\times D$
Therefore $A\times C$ is a subset of $B\times D$