For questions about relations that are reflexive, symmetric, and transitive. These are relations that model a sense of "equality" between elements of a set. Consider also using the (relation) tag.
An equivalence relation is a particular kind of relation that models a notion of "equality" between elements of a set. A relation $R$ on a set $X$ will be an equivalence relation if it satisfies the following properties:
- Reflexive – For each $a \in X$, we have $a \mathrel{R} a$.
- Symmetric – For any $a,b \in X$, $a \mathrel{R} b$ if and only if $b \mathrel{R} a$
- Transitive – For any $a,b,c \in X$, if $a \mathrel{R} b$ and $b \mathrel{R} c$, then $a \mathrel{R} c$.
Commonly the symbols $\equiv$ or $\cong$ or $\simeq$ or $=$ are used for equivalence relations instead of the letter $R$. Here are some examples of equivalence relations:
On the set $\mathbf{Z}$ of integers define the relation $\equiv_{37}$ on $\mathbf{Z}\times \mathbf{Z}$ by saying $a\equiv_{37} b$ if both $a$ and $b$ give the same remainder when divided by $37$. If $a \equiv_{37} b$ we say that $a$ and $b$ are congruent modulo $37$.
Let $T$ be the set of all triangles in the plane. An example of an equivalence relation on $T$ is the relation of two triangles being congruent.
