Transitive closure (relation) ($R^{+}={\bigcup }_{n\ge 1}{R}^n$)

The transitive closure of a relation connects $x$ to $y$ when a finite nonempty chain of $R$-steps leads from $x$ to $y$. Concatenation proves transitivity, and induction along a chain shows that any transitive relation containing $R$ contains $R^{+}$. This concerns closure of a relation, rather than the transitive closure of a set under membership.