Question

When is it possible to define the "irreflexive closure" of a relation \(R\), that is, a relation that contains \(R\), is irreflexive, and is contained in every irreflexive relation that contains \(R\)?

Short answer

This then also means that the irreflexive closure of a relation \(R\) is \(R\) itself, when \(R\) is irreflexive.

Step-by-step solution

Given

The given set is R.

Concept of Irreflexive

A relation\(R\)on a set\(A\)is irreflexive if\((a,a) \notin R\)for every element\(a \in A\).

Check the Set for Irreflexive

The irreflexive closure of a relation\(R\)cannot contain any elements of the form\((a,a) \notin R\)(by the definition of reflexive).

Since the irreflexive closure of relation\(R\)needs to contain the relation\(R\), the relation\(R\)can also not contain any elements of the form\((a,a) \notin R\).

By the definition of irreflexive, the relation\(R\)then has to be irreflexive itself.

This then also means that the irreflexive closure of a relation \(R\) is \(R\) itself, when \(R\) is irreflexive.