What do $R$, $R^2$ and $R^{-1}$ represent? Question
What do $R$, $R^2$ and $R^{-1}$ represent when $R$ is a relation on the set of natural numbers?
I'm doing some homework, but the $R^2$ and $R^{-1}$ notation confuses me.
Does $R^2 = R*R$?
Does $R^{-1} = \frac{1}{R}$?

A relation is a set of ordered pairs. You can imagine each ordered pair as an arrow: An arrow goes from $x$ to $y$ if $(x,y)\in R$.
With this visualization the inverse relation $R^{-1}$ just means reversing all arrows.
Composition of relations $S\circ R$ simply means that you start from one point and look where you can get if you follow an $R$-arrow first and then then an $S$-arrow.
For example have a look at this picture:

The pair $(0,2)$ belongs to the relation $S\circ R$, because there is a path from 0 to 2 (since you $(0,1)\in R$, and $(1,2)\in S$). Would you be able to find all ordered pairs belonging to $S\circ R$? (You may also notice that this is example of two equivalence relations such that their composition is not an equivalence relation.)
You have asked about $R\circ R$; it is a special case of this (when $S=R$).
See also: directed graph representing the inverse relation
