Why is the composition of relations $R$ and $S$ written $S \circ R$ instead of $R \circ S$?

Question

This is really basic (I'm new to this stuff), and doesn't even matter at all - But I'm just curious:

From my book:

If $R = (G,A,B)$ and $S = (H,B,C)$ the composition of $R$ and $S$ is known as $S \circ R = (H \circ G,A,C)$

First of all, if it says "of $R$ and $S$" I'd expect the $R$ to come first, and then the $S$. Which is not the case in $S \circ R$.

Even the graphs are inverted. I would expect $G$ to come first. But we got $(H \circ G,A,C)$.

I can live with that. But I'd like to know why is this the case. Why are they inverted?

Answer 1

Composition of relations is a generalization of composition of functions. When we have a function $f:A\to B$ and a function $g:B\to C$, the composite function is a function from $A$ to $C$ that takes $a$ to $g\big(f(a)\big)$, so it makes sense to write the composition in that same order: $g\circ f$ to match $g\big(f(a)\big)$ rather than $f\circ g$ to mean ‘apply $f$ first, then $g$’.

That said, however, it should be mentioned that some people do use the opposite convention for composition: when they write $f\circ g$, they do mean that $f$ is applied first, so that the result of applying $f\circ g$ to $a$ is $g\big(f(a)\big)$. When this convention is used, functions are often written to the right of their arguments: $(a)(f\circ g)=\big((a)f\big)g$. (The parentheses around the argument are often omitted.)

Answer 2

Because the argument of a function is usually written to its right. Then it looks like $$(S \circ R)(x) = S(R(x))$$ and function application is right to left.

Answer 3

It quite probably originates from function notation: If you have $f:A\to B$ and $g:B\to C$ and want to compose them, you get $g(f(x))$, so $g\circ f:A\to C$. This then got extended to relations and other things. I would personally prefer $f\circ g$ in this case and $xf$ or similar for function application.