Answer
$R_{1}=\{(a,x), (a,y)\}, R_{2}=\{(b,x), (b,y)\}, R_{3}=\{(a,y)\}, R_{4}=\{(b,y)\}$
Work Step by Step
Each relation must either (1) not use at least one element of $\{a,b\}$ as a first coordinate, or (2) map a single element of $\{a,b\}$ onto more than one element of $\{x,y\}$. Other possibilities include $R=\{(a,y)\}$, $R=\{(b,x)\}$, $R=\{(a,x), (a,y), (b,x), (b,y)\}$, $R=\{(a,x), (a,y), (b,x)\}$, $R=\{(a,x), (b,x), (b,y)\}$, and $R=\{\}$.
