Relation Properties

Some of the same-set relations we interact with have special properties:

• Definition. [Reflexive relations] A relation $R$ on a set $S$ is called reflexive if, for every $x \in S$, the pair $(x, x)$ is in $R$.
• Definition. [Anti-reflexive relations] A relation $R$ on a set $S$ is called anti-reflexive if, for every $x \in S$, the pair $(x, x)$ is NOT in $R$.
• Definition. [Symmetric relations] A relation $R$ on a set $S$ is called symmetric if, for every pair $(x, y)$ in $R$ (where $x, y \in S$,) the pair $(y, x)$ is also in $R$.
• Definition. [Anti-symmetric relations] A relation $R$ on a set $S$ is called anti-symmetric if the following is true: if the pairs $(x, y)$ and $(y, x)$ are both in $R$, then $x = y$.
• Definition. [Transitive relations] A relation $R$ on a set $S$ is called transitive if the following is true: if the pairs $(x, y)$ and $(y, z)$ are both in $R$, then the pair $(x, z)$ is also in $R$.

Clearly, some of these properties can't exist together for the same relation; for instance, no relation can be both reflexive and anti-reflexive.