Irreflexive (Antireflexive) Relations

What Is an Irreflexive Relation?

A relation on a set I is said to be irreflexive (also known as antireflexive) if no element of the set is related to itself. $$ \forall \ a \in I \ , \ a \require{cancel} \cancel{R} a $$

In an irreflexive relation, not a single element is connected to itself.

For example, the relation “A is the mother of B” is irreflexive, since no one can be their own mother.

Irreflexive (antireflexive) relations are a specific subset of relations defined on a set.

A Practical Example

Let’s take the finite set I:

$$ I = \{ 2,4,3,9,16 \} $$

We define a relation R where each element is linked to its square root, in other words: "x is the square root of y".

$$ R = \{ (4;2), (16;4), (9;3) \} $$

This relation is irreflexive because none of the elements in set I is the square root of itself.

Note. For instance, 2 is not the square root of 2. $$ 2 \ne \sqrt{2} $$ The same holds true for all other elements in I. In a graph representation, irreflexive relations have no loops - no node is connected to itself.

Further Remarks

Some important insights regarding irreflexive (antireflexive) relations:

• Some relations are neither reflexive nor irreflexive. So if a relation isn’t irreflexive, that doesn’t automatically mean it’s reflexive - and vice versa.

  Example. Consider the set $$ I = \{1,2,3,4,9,16 \} $$ and the relation R where "x is the square root of y". This relation is not reflexive since not all elements are related to themselves - for example, 2 is not the square root of 2. $$ 2 \ne \sqrt{2} $$ Yet the relation is not irreflexive either, because it includes at least one reflexive pair: for instance, 1 is the square root of 1. $$ 1 = \sqrt{1} $$ In the graph representing this relation, a loop appears on node 1. That means the relation fails to be irreflexive. However, since not all nodes have loops, it’s not reflexive either.
  

And so on.