How to prove this result about the binary relation?

Question

; defined as:

$R_1;R_2 = \{(a,c) : \text{there is a,b with } (a,b) ∈ R_1 \text{ and } (b,c) ∈ R_2\}$

I am not sure the proper method to prove this question.

I tried that

Ri+1 = Ri = Ri∪(R;Ri)

(R;Ri) ⊆Ri

then I don't know what to do next.

or could I assume that i=0?

Answer 1

This can be proven inductively. First you will need to establish your base that when $j=i$ that $R^i = R^j$.

Then show that for $j>i$ that if you assume $R^i = R^j$ you can prove $R^i = R^{j+1}$.