# Contents

## Definition

A (binary) relation $\sim$ on a set $A$ is irreflexive if no element of $A$ is related to itself:

$\forall (x: A),\; x \nsim x .$

(where $x\nsim x$ means that $x\sim x$ is false, i.e. that $\neg (x\sim x)$).

In the language of the $2$-poset Rel of sets and relations, a relation $R: A \to A$ is irreflexive if it is disjoint from the identity relation on $A$:

$\id_A \cap R \subseteq \empty .$

Of course, this containment is in fact an equality.

In constructive mathematics, if $A$ is equipped with a tight apartness $\#$, we say that $\sim$ is strongly irrelexive if only distinct elements of $A$ are related:

$\forall (x: A),\; \forall (y: A),\; x \sim y \;\Rightarrow x \# y .$

Since $\#$ is irrelexive itself, any strongly irrelexive relation must be irrelexive. The converse holds using excluded middle, through which every set has a unique tight apartness.

## Examples

• A digraph is a graph in which the edge relation is irreflexive.

Last revised on September 1, 2017 at 14:46:59. See the history of this page for a list of all contributions to it.