Given two sets A and $B,$ their cartesian product denoted as $AB$ is defined as follows:

$AB=\{(a,b)|ain{A}^{{?}^{?}}$binB$\}$

where $(a,b)$ denotes an ordered pair. For example, when $A=\{1,2\}$ and $B=\{2,3\}$ then we

have the set $AB=\{(1,2),(1,3),(2,2),(2,3)\}$ which has four elements.

$($a$)$ Suppose that $AB=\frac{O}{?}.$ What can we say about A and $B?$ Justify your answer.

$($b$)$ Suppose that $|AB|=13$ and What are the cardinalities of A and $B?$ Justify

your answer.