Relation Properties
Example: The $\text{LessThanOrEqualTo}$ ($\le$) relation on some set $S$ of comparable objects:
- It is reflexive because, for every $x \in S$, we indeed have $x \le x$, which means that $(x, x)$ is in $\text{LessThanOrEqualTo}$.
- Since it is reflexive, it is NOT anti-reflexive.
- It is NOT symmetric because $x \le y$ doesn't imply that $y \le x$. Example: since $2 \le 3$, we have $(2, 3) \in \text{LessThanOrEqualTo}$. However, it is clear that $3 \not\le 2$, so $(3, 2) \not\in \text{LessThanOrEqualTo}$.
- It is anti-symmetric because the only pairs $(x, y)$ and $(y, x)$ that exist in $\text{LessThanOrEqualTo}$ are those in which $x = y$. In other words, the relation $\text{LessThanOrEqualTo}$ contains pairs of the form $(x, x)$ for which we have $x \le x$.
- It is transitive since if $x \le y$ and $y \le z$, it must be that $x \le z$ (because $x \le y \le z$).
In summary, the $\text{LessThanOrEqualTo}$ relation has the following $3$ properties: reflexive, anti-symmetric, and transitive.
Definition. [Partial Order] A relation that is reflexive, anti-symmetric, and transitive is called a partial order.
These notes by Miriam Briskman are licensed under CC BY-NC 4.0 and based on sources.