Relation Properties

Example: A relation $\text{NotOddToEven}$ on some set $S$ of integers. In this relation, $(x, y) \in \text{NotOddToEven}$ if $x$ and $y$ are both even, or if $x$ and $y$ are both odd, or if $x$ is even and $y$ is odd (but not if $x$ is odd and $y$ is even.)

• It is reflexive because, for every $x \in S$, we indeed have $(x, x) \in \text{NotOddToEven}$ since $x$ can be either even (so we have an even-to-even pair) or odd (so we have an odd-to-odd pair.)
• Since it is reflexive, it is NOT anti-reflexive.
• It is NOT symmetric because we have $(2, 3) \in \text{NotOddToEven}$, but $(3, 2) \not\in \text{NotOddToEven}$ (since the odd-to-even pair $(3, 2)$ is not allowed.)
• It is NOT anti-symmetric because $(2, 4)$ and $(4, 2)$ are both in $\text{NotOddToEven}$, but $2 \neq 4$.
• It is transitive since $(x, y)$ and $(y, z)$ both being inside $\text{NotOddToEven}$ implies that $(x, z)$ is also in $\text{NotOddToEven}$. Fun In-Class Activity: Outline all the possible cases for $x$, $y$, and $z$ in terms of eveness and oddness to see that this relation is indeed transitive.

In summary, the $\text{NotOddToEven}$ relation has the following $2$ properties: reflexive and transitive.