Question

Let $A=\{1,2,3\},B=\{3,4\}$ and $C=\{4,5,6\}$. Find
$\left(A\right)A\times (B\cap C)$
$\left(B\right)(A\times B)\cap (A\times C)$
$\left(C\right)A\times (B\cup C)$
$\left(D\right)(A\times B)\cup (A\times C)$

Answer 1

Given: $A=\{1,2,3\},B=\{3,4\}$ and $C=\{4,5,6\}$

$B\cap C=\{3,4\}\cap \{4,5,6\}=\left\{4\right\}$
$A\times (B\cap C)=\{1,2,3\}\times \left\{4\right\}$
$=\left\{\right(1,4),(2,4),(3,4\left)\right\}$

$A\times B=\{1,2,3\}\times \{3,4\}$
$=\left\{\right(1,3),(2,3),(3,3),(1,4),(2,4),(3,4\left)\right\}$
$A\times C=\{1,2,3\}\times \{4,5,6\}$
$=\left\{\right(1,4),(2,4),(3,4),(1,5),(2,5),(3,5),(1,6),(2,6),(3,6\left)\right\}$
$(A\times B)\cap (A\times C)=$ set of common elements in both $A\times B$ and $A\times C$
$=\left\{\right(1,4),(2,4),(3,4\left)\right\}$

$B\cup C=\{3,4\}\cup \{4,5,6\}$
$=\{3,4,5,6\}$
$A\times (B\cup C)=\{1,2,3\}\times \{3,4,5,6\}$
$=\left\{\right(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,3),(3,4),(3,5),(3,6\left)\right\}$

$A\times B=\{1,2,3\}\times \{3,4\}$
$=\left\{\right(1,3),(2,3),(3,3),(1,4),(2,4),(3,4\left)\right\}$
$A\times C=\{1,2,3\}\times \{4,5,6\}$
$=\left\{\right(1,4),(2,4),(3,4),(1,5),(2,5),(3,5),(1,6),(2,6),(3,6\left)\right\}$
$(A\times B)\cup (A\times C)$
$=\left\{\right(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,3),(3,4),(3,5),(3,6\left)\right\}$