# 17.3: Properties of Relations

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Here we list some important properties a relation $$R$$ on a set $$A$$ can have.

## Reflexivity

##### Definition: Reflexive

$$a \mathrel{R} a$$ is true for all $$a \in A$$

## Symmetry and antisymmetry

##### Definition: Symmetric

for every pair of elements $$a_1,a_2 \in A$$ for which $$a_1 \mathrel{R} a_2$$ is true, $$a_2 \mathrel{R} a_1$$ is also true.

##### Remark $$\PageIndex{1}$$

The distinct part of the definition is important, since if $$a_1,a_2 \in A$$ are not distinct (i.e. $$a_2 = a_1$$), then obviously both $$a_1 \mathrel{R} a_2$$ and $$a_2 \mathrel{R} a_1$$ can be simultaneously true because they are the same statement.

##### Example $$\PageIndex{3}$$: An antisymmetric relation on real numbers.

The relation $$\mathord{\le}$$ on $$\mathbb{R}$$ is antisymmetric.

##### Example $$\PageIndex{4}$$: A relation can be neither antisymmetric nor symmetric.

On $$A = \{a,b,c\}\text{,}$$ the relation

\begin{equation*} R = \{(a,b),(b,a),(a,c)\} \subseteq A \times A \end{equation*}
is neither antisymmetric nor symmetric.

##### Example $$\PageIndex{5}$$: A relation can be both antisymmetric and symmetric.

The identity relation on any set, where each element is related to itself and only to itself, is both antisymmetric and symmetric.

##### Remark $$\PageIndex{2}$$

As Example $$\PageIndex{4}$$ and Example $$\PageIndex{5}$$ demonstrate, antisymmetry is not the opposite of symmetry. However, for a relation $$R$$ on set $$A\text{,}$$ we may think of symmetry and antisymmetry as being at opposite ends of a spectrum, measuring how often we have both $$a_1 \mathrel{R} a_2$$ and $$a_2 \mathrel{R} a_1$$ for $$a_1 \ne a_2\text{.}$$

By definition, antisymmetry is when we never have both. On the other hand, symmetry is when we always have both or neither; that is, for every distinct pair $$a_1,a_2 \in A\text{,}$$ we either have both $$a_1 \mathrel{R} a_2$$ and $$a_2 \mathrel{R} a_1\text{,}$$ or we have both $$a_1\ \not R\ a_2$$ and $$a_2\ \not R\ a_1\text{.}$$ However, a relation can fall between symmetry and antisymmetry on the spectrum, such as in Example $$\PageIndex{4}$$, where we sometimes have both (e.g. both $$a \mathrel{R} b$$ and $$b \mathrel{R} a$$ for that example relation) and we also sometimes have only one (e.g. $$a \mathrel{R} c$$ but $$c\ \not R\ a$$ for that example relation).

The equality relation on a set is a special case that is both symmetric and antisymmetric. In fact, equality is essentially the only relation that is both symmetric and antisymmetric — see Exercise 17.6.22.

In symbolic language, the definition of antisymmetric relation is

\begin{equation*} (\forall a_1 \in A)(\forall a_2 \in A)(a_1 \neq a_2 \Rightarrow a_1\ \not R\ a_2 \lor a_2\ \not R\ a_1 ) \text{.} \end{equation*}
However, in practise we usually prove antisymmetry using one of two logically equivalent formulations.

##### Test $$\PageIndex{3}$$: Antiymmetric relation.

To verify that relation $$R$$ on set $$A$$ is antisymmetric, prove either one of the following logical statements.

• $$(\forall a_1 \in A)(\forall a_2 \in A)(a_1 \neq a_2 \land a_1 \mathrel{R} a_2 \Rightarrow a_2\ \not R\ a_1)$$
• $$(\forall a_1 \in A)(\forall a_2 \in A)(a_1 \mathrel{R} a_2 \land a_2 \mathrel{R} a_1 \Rightarrow a_2 = a_1)$$
##### Remark $$\PageIndex{3}$$

The first formulation for proving antisymmetry provided above can be thought of as just a different way to say that it is not possible to have both $$a_1 \mathrel{R} a_2$$ and $$a_2 \mathrel{R} a_1$$ for distinct elements $$a_1,a_2\text{.}$$ The second formulation essentially says that the only possible way to have both $$a_1 \mathrel{R} a_2$$ and $$a_2 \mathrel{R} a_1$$ is if $$a_2 = a_1\text{.}$$

##### Note $$\PageIndex{1}$$

In Exercise 17.6.21 you are asked to prove that each of the two different ways of verifying that a relation is antisymmetric provided in the test above are equivalent.

## Transitivity

##### Definition: Transitive

for every triple of elements $$a_1,a_2,a_3 \in A$$ for which both $$a_1 \mathrel{R} a_2$$ and $$a_2 \mathrel{R} a_3$$ are true, $$a_1 \mathrel{R} a_3$$ must also be true.

This page titled 17.3: Properties of Relations is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform.