Question

Let $A=\left\{1,2\right\}$ and $B=\left\{3,4\right\}$. Write $A\times B$. How many subsets will $A\times B$ have? List them.

Answer 1

$A=\{1,2\}$ and $B=\{3,4\}$

$\therefore A\times B=\left\{\right(1,3),(1,4),(2,3),(2,4\left)\right\}$

$\Rightarrow n(A\times B)=4$

We know that if C is a set with $n\left(C\right)=m$, then $n\left[P\right(C\left)\right]=2^n$

Therefore, the set $A\times B$ has $2^4=16$ subsets. These are

$\varphi ,\left\{\right(1,3\left)\right\},\left\{\right(1,4\left)\right\},\left\{\right(2,3\left)\right\},\left\{\right(2,4\left)\right\},\left\{\right(1,3\left)\right(1,4\left)\right\},\left\{\right(1,3\left)\right\},\left\{\right(2,3\left)\right\},$

$\left\{\right(1,3),(2,4\left)\right\},\left\{\right(1,4\left)\right(2,3\left)\right\},\left\{\right(1,4\left)\right(2,4\left)\right\},\left\{\right(2,3\left)\right(2,4\left)\right\}$

$\left\{\right(1,3),(1,4),(2,3\left)\right\},\left\{\right(1,3),(1,4),(2,4\left)\right\},\left\{\right(1,3),(2,3),(2,4\left)\right\}$

$\left\{\right(1,4),(2,3),(2,4\left)\right\},\left\{\right(1,3),(1,4),(2,3),(2,4\left)\right\}$