Let R be a relation on a set X. Then R is transitive if and only if, for any finite sequence x0, x1,..., xn of elements...

Question

Proposition: Let R be a relation on a set X. Then R is transitive if and only if, for any finite sequence x0, x1,..., xn of elements of X such that xi−1 R xi for all i ∈ [n], we have x0 R xn.

This is a proposition proved in my textbook.

I understand the (=>) direction by proving with induction. Can someone please explain the (<=) direction in more detail? I'm not sure what this proof is saying. I read it as p(2) = x_{1}Rx_{2) means x_{0}Rx_{2}. But how does this show R is transitive? Thanks!

Answer 1

I read it as p(2) = x_{1}Rx_{2) means x_{0}Rx_{2}. But how does this show R is transitive?

Not quite $p(2)$ is the condition that

If $x_0 R x_1$ and $x_1 R x_2$ then $x_0 R x_2$.

Which is... the exact definition of transitivity.