- Two equivalent definitions of congruence mod m
- introduced equivalence classes

Notation:

*a*∣*b*is read "*a***divides***b*". By definition,*a*∣*b*if there is some*c*such that*c**a*=*b*.Definition: given an integer

*m*, two integers*a*and*b*are**congruent modulo m**if*m*∣(*a*−*b*). We write*a*≡*b*(*m**o**d**m*).Notation note: we are using that "mod" symbol in two different ways. The first was defined in a previous lecture:

*a**m**o**d**b*denotes the remainder when we divide*a*by*b*. The "mod m" in*a*≡*b*(*m**o**d**m*) is a note on the side of the equation indicating what we mean when we say " ≡ "Fact: These two uses of "mod" are quite related:

*a*≡*b*(*m**o**d**m*) if and only if*a**m**o**d**m*=*b**m**o**d**m*.Fact: for any

*m*, congruence mod*m*is an equivalence relation.

Given any equivalence relation ∼ on a set *A*, can break *A* up into several groups of interrelated elements.

Definition: if *x* ∈ *A*, then the **equivalence class of x** (written [

The set of all equivalence classes of elements of *A* is written ** A / ∼ ** and is pronounced "

Examples:

If

*A*is the set of people, and ∼ is the "is a (full) sibling of" relation, then*A*/ ∼ could be thought of as the set of (nuclear) families.If

*A*is the set of finite sets, and ∼ is the "has the same cardinality as" relation, then*A*/ ∼ has an element for each*n*∈ N.

We used the notion of equivalence classes to define the "modular numbers". The **set of integers mod m** (written Z

Example: - Z_{5} = {[0], [1], [2], [3], [4]} - Z_{3} = {[0], [1], [2]} = {{…, 6, − 3, 0, 3, 6, 9, …}, {…, − 5, − 2, 1, 4, 7, …}, {…, − 4, − 1, 2, 5, 8, …}}

In general, Z_{m} has *m* elements.

In future lectures, we will discuss how to add, multiply, exponentiate and divide equivalence classes.