I want to prove that $R^+ = R$ if $R$ is transitive.

Question

I want to prove that $R^+ = R$ if given $R$ is transitive.

I tried to prove it as follows:

$ R^+$

$ \bigcup\limits_{n=1}^\infty R^n$

It reduces as : $ R^n = R$ for all $n\ge 1$

I use Induction here

for $n=1$ its trival as $R^1=R$

for $n=n+1$ so $\qquad$ $R \circ R^n$

$ R \circ R$ $\qquad$ { I.H }

$R$ $\qquad$ { R is transitive }

Is my proof correct?

Answer 1

1. We don't necessarily have $R^n=R$ for a transitive relation in general. E.g. consider $<$ on $\{1,2,3,4\}$, here $R^4=\emptyset$.
2. As $R\subseteq R^+$, we only have to prove $\bigcup_{n\ge1}R^n\subseteq R$, that is, $R^n\subseteq R$ for all $n\ge 1$.
3. Yes, induction can work, and the hypothesis ($R$ being transitive) just translates to $R^2\subseteq R$.