# 7: Relations

• 7.1: Deﬁnition of Relations
A relation from a set A to a set B is a subset of A×B. Hence, a relation R consists of ordered pairs (a,b), where a∈A and b∈B. If (a,b)∈R, we say that is related to.
• 7.2: Properties of Relations
If R is a relation from A to A , then R⊆A×A ; we say that R is a relation on A .
• 7.3: Equivalence Relations
A relation on a set A is an equivalence relation if it is reflexive, symmetric, and transitive. We often use the tilde notation a∼b to denote an equivalence relation.
• 7.4: Partial and Total Ordering
Two special relations occur frequently in mathematics. Both have to do with some sort of ordering of the elements in a set. A branch of mathematics is devoted to their study. As you can tell from the brief discussion in this section, they cover many familiar concepts. A relation on a nonempty set A is called a partial ordering or a partial-order relation if it is reflexive, antisymmetric, and transitive. We often use ⪯ to denote a partial ordering, and called (A,⪯) a partially ordered set