# Metamath Proof Explorer

## Table of Contents - 5.8. Reflexive and transitive closures of relations

A relation, , has the reflexive property if holds whenever is an element which could be related by the relation, namely, an element of its domain or range. Eliminating dummy variables, we see that a segment of the identity relation must be a subset of the relation, or . See idref.

A relation, , has the transitive property if holds whenever there exists an intermediate value such that both and hold. This can be expressed without dummy variables as . See cotr.

The transitive closure of a relation, , is the smallest superset of the relation which has the transitive property. Likewise, the reflexive-transitive closure, , is the smallest superset which has both the reflexive and transitive properties.

Not to be confused with the transitive closure of a set, trcl, which is a closure relative to a different transitive property, df-tr.

1. The reflexive and transitive properties of relations
2. Basic properties of closures
3. Definitions and basic properties of transitive closures
4. Exponentiation of relations
5. Reflexive-transitive closure as an indexed union
6. Principle of transitive induction.