Elementary Foundations: An introduction to topics in discrete mathematics

Consider the relations $<$ and $=$ on $\mathbb{R}\text{,}$ and let $R$ be the union $<\cup =\text{.}$ Then $xRy$ means that at least one of $x<y$ or $x=y$ is true. That is, $R$ is the same as the relation $\le \text{.}$

Let $U$ be a universal set and consider the relation $\subseteq$ on $\mathcal{P}(U)\text{.}$ Then $A{\subseteq }^{\mathrm{c}}B$ means that $A$ is not a subset of $B\text{,}$ which can only happen if some elements of $A$ are not in $B\text{.}$ In other words, $A{\subseteq }^{\mathrm{c}}B$ means that $A\cap B^{\mathrm{c}}\ne \varnothing \text{.}$

Let $H$ represent the set of all living humans, and let $R$ represent the relation on $H$ where $h_{1}R{h}_2$ means human $h_1$ is the parent of human $h_2\text{.}$ Then $h_2{R}^{-1}{h}_1$ means human $h_2$ is the child of human $h_1\text{.}$ Both relations express the same information, but in a different order.

Recall that $\mid$ is a relation on ${\mathbb{N}}_{>0}$ where $m\mid n$ means that $m$ divides $n\text{.}$ Then for the inverse relation, $n{\mid }^{-1}m$ means $n$ is a multiple of $m\text{.}$ Both relations express the same information, but in a different order.

Let $\mathcal{L}$ represent the set of all possible logical statements. We have a relation $\equiv$ on $\mathcal{L}\text{,}$ where $A\equiv B$ means that logical statement $A$ involves the same statement variables and has the same truth table as logical statement $B\text{.}$ Since $A\equiv B$ if and only if $B\equiv A\text{,}$ we conclude that the logical equivalence relation on $\mathcal{L}$ is its own inverse.