Question

If $A=\{a,b\},B=\{x,y\}$ and $C=\{a,c,y\}$, then verify that $A\times (B\cap C)=(A\times B)\cap (A\times C)$

Answer 1

$B\cap C=\{x,y\}\cup \{a,c,y\}=\left\{y\right\}$

$A\times (B\cap C)=\{a,b\}\times \left\{y\right\}$

$=\{(a,y),(b,y)\}$

$\therefore A\times (B\cap C)=\{(a,y),(b,y)\}$ ..........$\left(1\right)$

$A\times B=\{a,b\}\times \{x,y\}=\{(a,x),(a,y),(b,x),(b,y)\}$

$A\times C=\{a,b\}\times \{a,c,y\}=\{(a,a),(a,c),(a,y),(b,a),(b,c),(b,y)\}$

$(A\times B)\cap (A\times C)=\{(a,x),(a,y),(b,x),(b,y)\}\cap \{(a,a),(a,c),(a,y),(b,a),(b,c),(b,y)\}$

$=\{(a,y),(b,y)\}$ .........$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$ we have

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

Hence verified.