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 $R^{-1} = \frac{1}{R}$?

$R^{-1}$ is the inverse relation: $R^{-1}=\{\langle \ell,k\rangle:\langle k,\ell\rangle\in R\}$. In other words, you get $R^{-1}$ from $R$ by turning each ordered pair in $R$ around. If $R=\{\langle n,n+1\rangle:n\in\Bbb N\}$, so that $R$ contains such pairs as $\langle 1,2\rangle$ and $\langle 7,8\rangle$, then $R^{-1}=\{\langle n+1,n\rangle:n\in\Bbb N\}$ and contains such pairs as $\langle 2,1\rangle$ and $\langle 8,7\rangle$.
$R^2$ is the composition of $R$ with itself. An ordered pair $\langle k,\ell\rangle$ is in $R^2$ if and only if there is an $n\in\Bbb N$ such that $\langle k,n\rangle$ and $\langle n,\ell\rangle$ are in $R$. In terms of the steppingstone analogy, if $R$ lets you get from $k$ to $\ell$ in two steps, then $R^2$ lets you do it in one. If $R$ is the relation that I used as an example in the first paragraph, $R$ lets you go from $k$ to $\ell$ in one step exactly when $\ell=k+1$, so it lets you get from $k$ to $\ell$ in two steps exactly when $\ell=k+2$. In this case, then, it turns out that
$$R^2=\{\langle n,n+2\rangle:n\in\Bbb N\}\;.$$
Note that if $R$ is transitive, and it lets you get from $k$ to $\ell$ in two steps, then by the definition of transitivity it already lets you get from $k$ to $\ell$ in one step. Thus, if $R$ is transitive you’ll always find that $R^2\subseteq R$: every pair in $R^2$ is already in $R$.
