Relations

In set theory, a relation $R$ (more specifically a binary relation) from the sets $S$ to $T$ is a subset of their Cartesian product $S\times T$. If $R\subseteq S\times T$ and if an ordered pair $(x,y)\in R$, then we write:

$$xRy$$

i.e., $x$ and $y$ are related by $R$. A relation from $S$ to $S$ is a relation in $S$.

This is super abstract. For a better idea of what this means, say $x\in S$ is in the set of husbands, and $y\in T$ is in the set of wives. $(x,y)\in R$ if $x$ is married to $y$, then we can say that $x$ and $y$ are related by saying that $x$ is married to $y$. Simple enough.

The inverse is a relation $R^{-1}$ from $T$ to $S$ defined by:

$$R^{-1}=\{(y,x)\in T\times S\mid (x,y)\in R\}$$

Properties

The relation $R\in S$ is:

• Reflexive if $(x,x)\in R$ for each $x\in S$.
• Irreflexive if $(x,x)\notin R$ for each $x\in S$.
• Symmetric if $(x_1,x_2)\in R$ implies that $(x_2,x_1)\in R$.
• Antisymmetric if $(x_1,x_2)\in R$ and $(x_2,x_1)\in R$ implies that $x_1=x_2$.