There's a very straight forward isomorphism between a set and it's indicator function. A relation \$$\triangle\$$ is defined to be a subset of \$$S \times S\$$. It's indicator function is then a function (\$$\chi)\$$ from a pair of elements of S to a Boolean value (True or False) where \$\chi (a,b) = True \iff a \triangle b \$

So technically what we get is the indicator function of a relation. But the distinction is entirely unimportant in normal mathematical reasoning.