Relations and their graphs

Definition. A binary relation $\mathcal{R}$ on $A$ is a subset $\mathcal{R}\subseteq A\times A$.

Example. $A=\{1,2,3\}$, $\mathcal{R}=\{(1,1),(1,2),(2,3)\}$ is a relation on $A$.

Notation. $(a,b)\in\mathcal{R}$ is often written $a\mathcal{R}b$.

Conclusion. Relations are sets of ordered pairs, allowing arbitrary patterns of connections.