Relation Properties

Example: The $\text{Equals}$ ($=$) relation on some set $S$ of comparable objects:

• It is reflexive because, for every $x \in S$, we indeed have $x = x$, which means that $(x, x)$ is in $\text{Equals}$.
• Since it is reflexive, it is NOT anti-reflexive.
• It is symmetric because $x = y$ indeed implies that $y = x$.
• It is anti-symmetric because the existence of the pairs $(x, y)$ and $(y, x)$ in $\text{Equals}$ means that $x = y$ (and, of course, that $y = x$.)
• It is transitive since if $x = y$ and $y = z$, it must be that $x = z$ (because $x = y = z$).

In summary, the $\text{Equals}$ relation has the following $4$ properties: reflexive, symmetric, anti-symmetric, and transitive.

Definition. [Equivalence Relation] A relation that is reflexive, symmetric, and transitive is called an equivalence relation.

The $\text{Equals}$ relation is the only anti-symmetric equivalence relation.