For a relation $R$, what does this notation $R^{\circ}R$, means?

Question

For a relation $R$, what does this notation $R^{\circ}R$, means?

Can anybody please help me in identifying this notation?

Actual question is: For the Relation $R=\{(1,2),(2,1),(2,3),(5,3),(4,5),(5,4),(5,5)\}$, find $R^{\circ}R$.

Answer 1

$R\circ R=\{(a, b): \mbox{ for some $c$, $aRc$ and $cRb$}.\}$

It's the relation version of composition. To see this, suppose I have a function $f$. I can represent $f$ by its graph, the relation $R_f=\{(a, b): f(a)=b\}$.

Then the graph of the composition, $R_{f\circ f}$, is $$R_{f\circ f}=\{(a, b): f(f(a))=b\}=\{(a, b): \exists c(aR_fc\mbox{ and }cR_fb)\}=R_f\circ R_f$$ (namely, take $c=f(a)$).

Since relations can be much more complicated than functions, though, compositions of relations can behave weirdly. However, some basic properties still hold. In particular, composition of relations is associative: $R_1\circ (R_2\circ R_3)=(R_1\circ R_2)\circ R_3$.