Binary Relations

Binary relation

Take two arbitrary sets, $A$ and $B$. Remember that the Cartesian product $A\times B$ is a set containing ordered pairs $(a,b)$ so that $a\in A$ and $b\in B$. Any subset $R$ of $A\times B$ is a relation from $A$ to $B$. We call this a binary relation because it’s defined for two sets. We’ll simply use the term “relation” for binary relations unless mentioned otherwise.

Examples

Let $A=\{1,2,3,4\}$ and $B=\{a,b,c\}$.

Here are some examples of binary relations from $A$ to $B$:

Given a relation $R$, if $(a,b) \in R$, we say that the element $a$ is related to the element $b$ through the relation $R$. We denote this as $aRb$. For example, the element $1$ of the set $A$ is related to the element $a$ of the set $B$ through the relation $R1$. We write this information as $1R_1a$. Similarly, we should note that $4R_1a$, $3R_2c$ ...