Notation: a∣b is read "a divides b". By definition, a∣b if there is some c such that ca = b.
Definition: given an integer m, two integers a and b are congruent modulo m if m∣(a − b). We write a ≡ b (mod m).
Notation note: we are using that "mod" symbol in two different ways. The first was defined in a previous lecture: a mod b denotes the remainder when we divide a by b. The "mod m" in a ≡ b (mod 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 (mod m) if and only if a mod m = b mod 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 [x] ∼ ) is the set of all elements of A that are related to x. If ∼ is clear from context I will often omit it and just write [x].
The set of all equivalence classes of elements of A is written A / ∼ and is pronounced "A mod ∼ ".
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 Zm) is the set of all integers modulo the "congruent mod m" relation.
Example: - Z5 = {[0], [1], [2], [3], [4]} - Z3 = {[0], [1], [2]} = {{…, 6, − 3, 0, 3, 6, 9, …}, {…, − 5, − 2, 1, 4, 7, …}, {…, − 4, − 1, 2, 5, 8, …}}
In general, Zm has m elements.
In future lectures, we will discuss how to add, multiply, exponentiate and divide equivalence classes.