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Congruence modulo every integer is an equivalence relation on $Z$

Statement

For every integer $n$ , the relation $a \equiv b (m o d n)$ on $Z$ (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ) is an equivalence relation (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ). For $n = 0$ it is equality, and for $n = 1$ it is the universal relation.

Facts & Assumptions

Given: An integer $n$ and integers $a, b, c$ .

[F1]

$a \equiv b (m o d n)$ means $n ∣ (a - b)$ (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

[L1]

Divisibility is reflexive and linear: $d ∣ u$ and $d ∣ v$ imply $d ∣ u x + v y$ for all integers $x, y$ ; also $d ∣ u$ implies $d ∣ - u$ (Divisibility is reflexive and transitive on $Z$ , and is linear: if $d ∣ a$ and $d ∣ b$ then $d ∣ a x + b y$ for all integers $x, y$ ; also $d ∣ a$ implies $d ∣ a c$ , $- d ∣ a$ and $d ∣ - a$ ).

[F2]

An equivalence relation is reflexive, symmetric and transitive (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

Proof

technique · direct

1.1

Reflexivity: $a - a = 0 = n \cdot 0$ , so $n ∣ (a - a)$ and $a \equiv a (m o d n)$ .

F1algebra

1.2

Symmetry: if $a \equiv b (m o d n)$ , then $n ∣ (a - b)$ , hence $n ∣ - (a - b) = b - a$ , so $b \equiv a (m o d n)$ .

F1L1

1.3

Transitivity: if $a \equiv b (m o d n)$ and $b \equiv c (m o d n)$ , then $n ∣ (a - b)$ and $n ∣ (b - c)$ , hence $n ∣ (a - b) + (b - c) = a - c$ , so $a \equiv c (m o d n)$ .

F1L1algebra

2.1

Steps 1.1, 1.2 and 1.3 establish the three clauses in [F2], so congruence modulo $n$ is an equivalence relation.

step 1.1step 1.2step 1.3F2

3.1

When $n = 0$ , [F1] reads $0 ∣ (a - b)$ , which is equivalent to $a = b$ ; when $n = 1$ , it always holds because $a - b = 1 \cdot (a - b)$ .

F1algebra∎

Depends on

Used by

The congruence class [a]ₙ and the quotient set ℤ/n Definition

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Conrad, Modular Arithmetic (standard reference, not scraped)

Congruence modulo every integer is an equivalence relation on ZZ