Let A = {a, b}. List all relation on A and find their number

Any relation on A is a subset of A×A.

$A \times A=\{(a, a),(a, b),(b, a),(b, b)\}$

The subsets are.

{} empty set

$\{(a, a)\}$

$\{(a, b)\}$

$\{(a, a),(a, b)\}$

$\{(b, a)\}$

$\{(b, b)\}$

$\{(b, a),(b, b)\}$

$\{(a, a),(b, a)\}$

$\{(a, b),(b, a)\}$

$\{(a, a),(b, a),(b, b)\}$

$\{(a, a),(b, b)\}$

$\{(a, a),(a, b),(b, a)\}$

$\{(a, a),(a, b),(b, b)\}$

$\{(a, b),(b, a),(b, b)\}$

$\{(a, a),(a, b),(b, a),(b, b)\}$

Thus, there are 16 total relations.