Properties of relations on a set

Naive set theory: Relations

Properties of relations on a set

Properties of a binary relation

Properties

A binary relation $R$ on a set $X$ is called:

reflexive if $xRx$ for every $x\in X$ ;
symmetric if for all $x,y\in X$ : if $xRy$ , then $yRx$ ;
antisymmetric if for all $x,y\in X$ : if $xRy$ and $yRx$ , then $x=y$ ;
asymmetric if there is no pair $(x,y)\in X\times X$ bsuch that $xRy$ and $yRx$ ;
transitive if for all $x,y,z\in X$ : if $xRy$ and $yRz$ , then $xRz$ .

Examples

The relation $\le$ on $\mathbb{Z}$ is reflexive, not symmetric, antisymmetric, and transitive.

The relation $\lt$ on $\mathbb{Z}$ is not reflexive, not symmetric, assymetric, and transitive.

The relation $=$ on $\mathbb{Z}$ is reflexive, symmetric, antisymmetric, and transitive.

The relation "is parallel to" on the set of lines in the plane is reflexive, symmetric, and transitive.

Let $R$ be a relation on a set $X$ . Then:

$R$ is reflexive if and only if $\mathrm{Id}(X)=\{(x,x)\mid x\in X\}\subset R\cap R^{-1}$ ;
$R$ is symmetric if and only if $R^{-1}=R$ ;
$R$ is antisymmetrisch dan en slechts dan als $R\cap R^{-1}\subset \mathrm{Id}_X=\bigl\{(x,x)\mid x\in X\bigr\}$ ;
$R$ is transitive if and only if $R\circ R\subset R$ .

The proof is in fact nothing more than another way of writing down what the definitions of the properties reflexive, symmetric, antisymmetric and transitive for a relation are.