Relations

What is it?

In Discrete Mathematics, relations are used to evidence some sort of relationship between two sets. Given two non-empty sets $A$ and $B$, the relation $R$ from $A$ to $B$ is a subset of $A\times B$. Then, the relation $R$ consists of ordered pairs $(a,b)$, where $a\in A$ and $b\in B$. If $(a,b)\in R$, it’s considered to be related to. However, $a$ being related to $b$ does not necessarily imply that $b$ is related to $a$.

A relation $R$, by itself, can be considered as a kind of ‘rule’ between $a$ and $b$. If $a,b$ are related by $R$, that means they obey the same rule $R$, and can be written as $xRy$.