Let A = {1, 2} and B = {3, 4}. Write A × B.
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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