I am working on some discrete mathematics and came across this strange operator on two sets.

$R \circ S$

I have only seen this circle operator with function compositions, so is this "Set Composition"? If so, then how does it work?

I found this on WebWork. The question is

"Suppose R and S are relations on a set A. If R and S are reflexive
relations, then R $\circ$ S is reflexive" select true or false.

  • 3
    $\begingroup$ Composition of relations. $\endgroup$ – user61527 Feb 13 '14 at 4:43
  • $\begingroup$ Yes, composition of relations, which in the special case when the relations are functions, is exactly the same as composition of functions. $\endgroup$ – ShreevatsaR Feb 13 '14 at 4:50

When $R$ is a relation on sets $G$ and $H$ (that is, a subset of $G\times H$), and $S$ is a relation on sets $H$ and $J$, then $S\circ R$ is a relation on $G$ and $J$ in which $g$ is related to $j$ if and only if there is some $h\in H$ with $g R h$ and $h S j$.

For example, suppose $(g,h)\in R$ means that woman $g$ is the mother of person $h$, and $(h, j)\in S$ means that person $h$ likes to eat food $j$. Then $S\circ R$ is the relation which holds for woman $g$ and food $j$ if and only if $g$ is the mother of someone who likes to eat food $j$.

When $R$ and $S$ are functions, this definition coincides with the composition of the two functions.


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