# Thread: composition of relations

1. ## composition of relations

Hello, I'm having problems undestanding an example. It's supposed to be simple, but I can't quite draw the conclusions.

Let there be a Set, $\displaystyle M$ and let there be $\displaystyle Map(M, M) := \{f~|~ f: M \rightarrow M$ is a mapping$\displaystyle \}$. If $\displaystyle f, g \in Map(M, M)$, so is $\displaystyle f \circ g$ also in $\displaystyle Map(M, M)$. And so is $\displaystyle \circ$ a composition of $\displaystyle Map(M, M)$.

My question is, what exactly is $\displaystyle Map(M, M)$? Is it the set $\displaystyle M$x$\displaystyle M$? And are $\displaystyle f$ and $\displaystyle g$ therefore $\displaystyle Map(f, f')$ and $\displaystyle Map(g, g')$? Because, in the set definition $\displaystyle Map(M, M) := \{f~|~ f: M \rightarrow M\}$, you have a mapping from $\displaystyle M \rightarrow M$?

2. Oh! This is about the image of a function?! And M is the domain, and codomain of the two functions f and g?! Still, if that's right, it's hard to understand from the example. So the example is saying that a composition exists for any two functions if they share the same domain and codomain.

3. I *think* that this example was a prelude to an introduction to matrixes.