The composition of the $<$ relation with itself

Question

I am struggle with answering this question. I do not understand how to approach this question.

1.Let $<$ denote the less than relation on the set of integers. Describe the squared relation $<^{2}$. Is it the same as $<$ ?

Answer 1

We assume that "squared relation" means the composition (or relative product) of $<$ with itself, where the relative product of two relations $R,S$ is the relation :

$Z = \{ \langle x,y \rangle : \exists z(xRz \land zSy) \}$.

then we have to apply the definition with $<$ in place of both $R$ and $S$.

Thus :

$<^2 = \{ \langle x,y \rangle : \exists z(x < z \land z < y) \}$.

Thus, on the set of integers $\langle n,n+1 \rangle \in <$, while $\langle n,n+1 \rangle \notin <^2$.