A **binary relation** is a collection of (ordered) pairs (a,b) of [[Element of a Set|elements]] from the same [[Set|set]]. Given a set $A$, a binary relation on $A$ is a subset $\mathscr{R} \subset A \times A$, which are those pairs where the relation is said to hold. Instead of $(a, b) \in \mathscr{R}$, we write $a \mathscr{R} b$. Examples of **binary relations** Membership (Element of): $3\in\{1,3,5\}$ Equality and Inequality: $2<3$ is equivalent to saying $(2,3)\in\{(a,b)\mid b-a \text{ is positive}\}$