Question:
Let $A=\{1,2\}$ and $B=\{3,4\}$. Write $A \times B$. How many subsets will $A \times B$ have? List them.
Solution:
$A=\{1,2\}$ and $B=\{3,4\}$
$\therefore A \times B=\{(1,3),(1,4),(2,3),(2,4)\}$
$\Rightarrow n(A \times B)=4$
We know that if $C$ is a set with $n(C)=m$, then $n[P(C)]=2^{m}$.
Therefore, the set $A \times B$ has $2^{4}=16$ subsets. These are
$\Phi,\{(1,3)\},\{(1,4)\},\{(2,3)\},\{(2,4)\},\{(1,3),(1,4)\},\{(1,3),(2,3)\}$
$\{(1,3),(2,4)\},\{(1,4),(2,3)\},\{(1,4),(2,4)\},\{(2,3),(2,4)\}$
$\{(1,3),(1,4),(2,3)\},\{(1,3),(1,4),(2,4)\},\{(1,3),(2,3),(2,4)\}$,
$\{(1,4),(2,3),(2,4)\},\{(1,3),(1,4),(2,3),(2,4)\}$
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