Composite relation of R

Question

I'm new to relations in discrete mathematics and I am having a hard time understanding how to do this exercise.

Let $R$ be a binary relation on the set of integers such that $(a,b) \in R$ if and only if $b-a=1$. What is the composite relation $R \circ R$?

Answer 1

The composite relation $R \circ R$ consists of those pairs $(a,c)$ such that there is a $b$ where $(a,b)$ and $(b,c)$ are both members of $R$. Suppose I give you a number $a$. How many pairs $(a,b) \in R?$

Answer 2

By definition, $a\mathrel{(R\circ R)}b$ iff there exists $c$ with $a\mathrel R c$ and $c\mathrel R b$, The latter means $c-a=1$ and $b-c=1$, hence $b-a=2$. We conclude that $a\mathrel{(R\circ R)} b$ iff $b-a=2$ - at least provided that in our underlying set, we always have $a+1$ in int whenever $a$ and $a+2$ are in it!