Equivalence Relations

Equivalence relation

A relation $R$ is an equivalence relation on set $A$ if it has the following three properties:

• $R$ is reflexive, that is, for every element $a$ of $A$, we have $(a,a)$ in $R$.

• $R$ is symmetric, that is, if $(a,b)$ is in $R$, then $(b,a)$ is also in $R$.

• $R$ is transitive, that is, if $(a,b)$ and $(b,c)$ are in $R$ , then $(a,c)$ is also in $R$.

By definition, this means that an equivalence relation has to be reflexive, symmetric, and transitive. Let’s explore a few examples to help us understand this concept further.

Examples

Let’s consider $L$—the set of lines in the Euclidean plane $\mathbb{R}^2$—and the relation of parallelism on it. Two lines are parallel to each other if they have the same slope, as shown below: