
# 7: Equivalence Relations

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In Section 6.1, we introduced the formal definition of a function from one set to another set. The notion of a function can be thought of as one way of relating the elements of one set with those of another set (or the same set). A function is a special type of relation in the sense that each element of the first set, the domain, is “related” to exactly one element of the second set, the codomain. This idea of relating the elements of one set to those of another set using ordered pairs is not restricted to functions. For example, we may say that one integer, a , is related to another integer, b , provided that a is congruent to b modulo 3. Notice that this relation of congruence modulo 3 provides a way of relating one integer to another integer. However, in this case, an integer a is related to more than one other integer.

• 7.1: Relations
The notion of a function can be thought of as one way of relating the elements of one set with those of another set (or the same set). A function is a special type of relation in the sense that each element of the first set, the domain, is “related” to exactly one element of the second set, the codomain. This idea of relating the elements of one set to those of another set using ordered pairs is not restricted to functions.
• 7.2: Equivalence Relations
An equivalence relation on a set is a relation with a certain combination of properties that allow us to sort the elements of the set into certain classes. Let A be a nonempty set. A relation ∼ on the set A is an equivalence relation provided that ∼ is reflexive, symmetric, and transitive. For a,b∈A , if ∼ is an equivalence relation on A and a ∼ b , we say that a is equivalent to b. In this section, we will focus on the properties that define an equivalence relation.
• 7.3: Equivalence Classes
An equivalence relation on a set is a relation with a certain combination of properties (reflexive, symmetric, and transitive) that allow us to sort the elements of the set into certain classes.
• 7.4: Modular Arithmetic
The term modular arithmetic is used to refer to the operations of addition and multiplication of congruence classes in the integers modulo n.
• 7.S: Equivalence Relations (Summary)