Let $R$ be the relation on $\Bbb Z$ such that $xRy$ iff $x-y=c$
a.) Define $R^2$
b.) Define $R^i$ for abitrary $i\ge1$.
Well the problem I'm having with this is trying to figure out how get the power of a relation and how does $xR^2y$ for example. Well I know for starters that in the case with $R^2$ we have $R \cdot R$. Any input would be great.
firstly think of a relation on a finite set $A$ with labelled elements $a_1, a_2, \dots, a_n$
the relation $R$ on $A$ is a subset of $A \times A$, so it can be represented by an $n \times n$ matrix $(r_{ij})$ whose elements take the value $0$ or $1$. $r_{ij}=1$ if $a_iRa_j$ and $0$ otherwise (c.f. the indicator function of a set).
the relation $R^2$ is defined by: $$ a_iR^2a_k \equiv \exists j.a_iRa_j\land a_jRa_k $$ thus $R^2$ can be computed by an operation very similar to matrix multiplication of $R$ by itself, except that we take the boolean union rather than the arithmetic sum, so that the matrix corresponding to $R^2$ has elements $$r_{ij}^2 = \lor_k \;( r_{ik} \land r_{kj})$$
for $\mathbb{Z}$ it is easy to extend this idea to use an infinite matrix.
you can probably see that the matrix for $R$ on $\mathbb{Z}$ to be the identity matrix but shifted to the left by $c$ places. (when i wrote this i was assuming $c$ to be positive. of course if $c$ is negative we must shift to the right)
when multiplying this matrix by itself we find this shifts it another $c$ places to the left (or right for negative $c$, which means that: $$ xR^2y \equiv x - y = 2c $$
if at first you have difficulty visualizing this, draw out the first few entries taking a small value of $c$ and do the "multiplication" - the underlying principle will soon become clear, and this should enable you to solve the second part of the question.
i would have chosen a superior notation if i had used the symbol $R_c$ for the original relation. with this notation you can investigate the product relation $R_cR_d$. you should be able to see that the relations thus defined form a (multiplicative) abelian group isomorphic to the additive group of $\mathbb{Z}$