Binary Relation

Definition

A binary relation on a given set $A$ is a set of ordered pairs that is a subset of $A\times A$. For an ordered pair $(x,y)\in R$ in a binary relation $R$, there is a specific relationship between $x$ and $y$. This is denoted as $xRy$, $R(x,y)$, $x{\sim }_{R}y$, or $x\sim y$.

Example

Typical examples of binary relations include reflexive, symmetric, and transitive relations.

1. Reflexive: $a\sim a$
2. Symmetric: $a\sim b$ if and only if $b\sim a$
3. Transitive: $a\sim b$ and $b\sim c$ imply $a\sim c$
4. Anti-symmetric: $a\sim b$ and $b\sim a$ imply $a=b$