# Find the domain and image of the relation $R=\{(a, b), (c, b), (a, b)\}$

Let $A={a, b, c}$, and let $R=\{(a, b), (c, b), (a, b)\}$. Find the domain of $R$ and the image of $R$.

This would be very elementary, but I want to get my answer checked.

Let $R$ be a relation from $A$ to $B$. Then by definition, then the domain of the relation $R$ in symbols is $$\operatorname{Dom}(R)=\{a\in A\;\vert\;(a, b)\in R\text{ for some }b∈B\}$$

and the image of the relation $R$ in symbols is $$\operatorname{Im}(R)=\{b\in \;\vert\;(a, b)\in R\text{ for some }a\in A\}$$

So, $\operatorname{Dom}(R)=\{a, c\}, \operatorname{Im}(R)=\{b\}$.

$Dom(R)$=$\{a,c\}$ and $Image(R)$=$\{b\}$
Because in the $2$ distinct ordered pairs in $R$, the pre-image parts are $a$ and $c$ whereas the image is always $b$.