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.

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