Equivalence Relation

Topics: Discrete Mathematics - Algebra - Relation

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(definition)

A relation in a set $A$ is an equivalence relation if it satisfies the following properties:

• It’s reflexive
• It’s symmetric
• It’s transitive

We denote an equivalence relation with $\sim$.

Example

Let $A=\mathbb{R}$. Let $R$ be defined by:

$$xRy⟺x^2=y^2$$

$R$ is…

• Reflexive, since $x^2=x^2$
• Symmetric, since if $x^2=y^2$, then $y^2=x^2$
• Transitive, since if $x^2=y^2$ and $y^2=z^2$, then $x^2=z^2$

Therefore, $R$ is an equivalence relation