Let A=

Any relation in A can be written as a set of ordered pairs. 

The only ordered pairs that can be included are (a, a), (a, b), (b, a) and (b, b). 

There are four ordered pairs in the set, and each subset is a unique combination of them. 

Each unique combination makes different relations in A. 

{ } [the empty set] 

{(a, a)} 

{(a, b)} 

{(a, a), (a, b)} 

{(b, a)} 

{(a, a), (b, a)} 

{(a, b), (b, a)} 

{(a, a), (a, b), (b, a)} 

{(b, b)} 

{(a, a), (b, b)} 

{(a, b), (b, b)} 

{(a, a), (a, b), (b, b)} 

{(b, a), (b, b)} 

{(a, a), (b, a), (b, b)} 

{(a, b), (b, a), (b, b)}

{(a ,a), (a, b), (b, a), (b, b)}

Number of elements in the Cartesian product of $A$ and $A=2 \times 2=4$

$\therefore$ Number of relations $=2^{4}=16$