Question

If $A=\{1,2\}$, $B=\{1,2,3\}$, $C=\{1,5,7\}$, $D=\{2,4,7\}$, Show that

(i) $(A\times B)\cap (C\times D)=(A\cap C)\times (B\cap D)$
(ii) $(A\times B)\cup (C\times D)\le (A\cup C)\times (B\cup D)$

Answer 1

$A\times B=\left\{\right(1,1),(1,2),(1,3),(2,1),(2,2),(2,3\left)\right\}$
$C\times D=\left\{\right(1,2),(1,4),(1,7),(5,2),(5,4),(5,7),(7,2),(7,4),(7,7\left)\right\}$

(i)

$(A\times B)\cap (C\times D)=\left\{\right(1,2\left)\right\}$ ......(1)

$A\cap C=\left\{1\right\}$
$B\cap D=\left\{2\right\}$
$(A\cap C)\times (B\cap D)=\left\{\right(1,2\left)\right\}$ ...(2)

from (1) and (2)
$(A\cap B)\times (C\cap D)=(A\cap C)\times (B\cap D)$

(ii)$(A\times B)\cup (C\times D)=\left\{\right(1,1),(1,2),(1,3),(1,4),(1,7),(2,1),(2,2),(2,3),(5,2),(5,4),(5,7),(7,2),(7,4),(7,7\left)\right\}$

$(A\cup C)=\{1,2,5,7\}$

$(B\cup D)=\{1,2,3,4,7\}$

$(A\cup C)\times (B\cup D)=\left\{\right(1,1),(1,2),(1,3),(1,4),(1,7),(2,1),(2,2),(2,3),(2,4),(2,7),(5,1),(5,2),(5,3),(5,4),(5,7),(7,1),(7,2),(7,3),(7,4),(7,7)\}$

$\therefore (A\times B)\cup (C\times D)\le (A\cup C)\times (B\cup D)$