# Relations

This page is a sub-page of our page on Some basic algebraic concepts.

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Relation: A relation on a set $\mathcal X$ is a subset $R \subseteq \mathcal X \times \mathcal X$.
If $(x, y) \in R$ we say that $x$ is related to $y$ and we write $xRy$.

A relation $R$ on $\mathcal X$ is called:

reflexive if $xRx \, \forall x \in \mathcal X$.

symmetric if $xRy \Rightarrow yRx \, , \forall x, y \in \mathcal X$.

antisymmetric if $xRy \, \text{and} \, yRx \Rightarrow x=y \, , \forall x, y \in \mathcal X$.

transitive if $xRy \, \text{and} \, yRz \Rightarrow xRz \, , \forall x, y,z \in \mathcal X$.

Four different types of order relations:

Pre order (relation): A reflexive and transitive relation.

Partial order (relation): A reflexive, antisymmetric and transitive relation.

Total order (relation): A partial order such that $xRy$ or $yRx$ holds $\forall x, y \in \mathcal X$.

Equivalence relation: A reflexive, symmetric, and transitive relation.