Question

Given $A=1,2,3,B=3,4,C=4,5,6,$
then show that $A\cup (B\cup C)=\{1,2,3,4,5,6\}$
and $(A\times B)\cap (B\times C)=\{3,4\}$

Answer 1

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