The Timpson Lyceum · Discrete Math (2025)

Equivalence Relations and Partitions - Lecture 8 (Discrete Math)

Equivalence Relations and Partitions – Lecture 8

Overview

This lecture pulls together the three "nice" properties a relation can have — reflexive, symmetric, and transitive — into a single named object: the equivalence relation. We see why these relations are so common (congruence, similarity, "same height," "same last name" all qualify) and study the most important arithmetic example, congruence modulo $n$. From there we build the equivalence class of an element and prove the central structural fact: equivalence classes are nonempty, pairwise disjoint, and cover the whole set — i.e., they form a partition. Conversely, any partition induces an equivalence relation. We close by using equivalence classes as a counting tool to count arrangements of letters with repeats (e.g., the letters of "Mississippi").

Key Concepts

Equivalence relation. Let $R$ be a relation on a set $A$. $R$ is an equivalence relation if it is:

• Reflexive — for all $x \in A$, $x \mathrel{R} x$ (everything is related to itself).
• Symmetric — for all $x, y \in A$, if $x \mathrel{R} y$ then $y \mathrel{R} x$.
• Transitive — for all $x, y, z \in A$, if $x \mathrel{R} y$ and $y \mathrel{R} z$ then $x \mathrel{R} z$.

An equivalence relation is a precise way of saying "two objects are, in some sense, the same." Familiar examples: same height, same last name, geometric congruence, and similarity.

Congruence modulo $n$. Let $n$ be a positive integer. For integers $x$ and $y$ we say $x$ is congruent to $y$ modulo $n$, written $x \equiv y \pmod{n},$ provided $n \mid (x - y)$ — that is, the difference $x - y$ is divisible by $n$. By definition $n \mid (x-y)$ means there exists an integer $k$ with $nk = x - y$.

• $13 \equiv 8 \pmod 5$ is true since $13 - 8 = 5$ and $5 \mid 5$; but $13 \equiv 7 \pmod 5$ is false since $5 \nmid 6$.
• $0 \equiv 8 \pmod 2$ is true since $0 - 8 = -8$ and $2 \mid -8$ (it divides four times — a negative result still counts).
• $7 \equiv 7 \pmod 9$ is true since $7 - 7 = 0$ and $n \mid 0$ for every $n$ (zero times).
• Any odd integer is $\equiv$ any other odd integer $\pmod 2$, because odd $-$ odd is even.

Congruence mod $n$ is an equivalence relation (a key theorem of this lecture; transitivity is left as an exercise):

• Reflexive: $x - x = 0$ and $n \mid 0$, so $x \equiv x \pmod n$.
• Symmetric: if $x \equiv y$, then $nk = x - y$ for some integer $k$; multiplying by $-1$ gives $n(-k) = y - x$, so $n \mid (y - x)$ and $y \equiv x \pmod n$.

Equivalence class. Let $R$ be an equivalence relation on $A$ and let $a \in A$. The equivalence class of $a$ is the set of all elements related to $a$: $[a] = \{\, x \in A : x \mathrel{R} a \,\}.$

• Under same last name on a room of people, $[a]$ is the set of everyone sharing $a$'s last name.
• Under congruence mod $2$, $[1] = \{\dots, -3, -1, 1, 3, 5, \dots\}$ = all odd integers, and $[0]$ = all even integers. Note $[1] = [-117]$: the relation splits $\mathbb{Z}$ into just two classes, the evens and the odds.

Classes are equal exactly when their reps are related. For $a, b \in A$: $a \mathrel{R} b \iff [a] = [b].$ Consequently, if two equivalence classes overlap at all, they are identical — if $[a] \cap [b] \neq \varnothing$, then $[a] = [b]$.

Partition. A partition of $A$ is a set $P$ of nonempty, pairwise-disjoint subsets of $A$ (called parts) whose union is all of $A$. No part may be empty, no element may sit in two different parts, and together the parts recover $A$.

The fundamental correspondence (equivalence relations $\leftrightarrow$ partitions):

• The equivalence classes of any equivalence relation on $A$ form a partition of $A$. They are nonempty (each $[a]$ contains $a$ by reflexivity), pairwise disjoint (overlapping classes are equal), and their union is $A$ (every element lies in its own class).
• Conversely, given a partition $P$, the same-part relation — $a \mathrel{R} b$ iff some part of $P$ contains both $a$ and $b$ — is an equivalence relation. Reflexivity follows because the parts cover $A$; symmetry because membership in a set is unordered; transitivity because two parts sharing the common element $b$ must be the same part (pairwise disjoint).

Counting with equivalence classes. To count distinct arrangements when some objects are interchangeable, count all arrangements as if everything were distinct, then divide by the size of each equivalence class (the number of indistinguishable rearrangements that produce the same outcome): $\#(\text{distinct arrangements}) = \frac{\text{total arrangements}}{\text{size of each class}}.$

Worked Example

Count the distinct arrangements of the letters of "MISSISSIPPI."

The word has $11$ letters with repeats: one $\text{M}$, four $\text{I}$'s, four $\text{S}$'s, and two $\text{P}$'s. Treating all $11$ tiles as distinct gives $11!$ orderings, but swapping identical letters among themselves yields the same word. Group orderings into equivalence classes ("spell the same word"); each class has size $4! \cdot 4! \cdot 2! \cdot 1!$ (the rearrangements of the I's, S's, P's, and the lone M). Divide total by class size:

(The simpler "WORD" version: $4$ distinct tiles give $4! = 24$ words; if one letter were doubled you'd divide by $2! = 2$.)

Summary

• An equivalence relation is a relation that is reflexive, symmetric, and transitive; it formalizes "being the same in some respect."
• Congruence mod $n$ ($x \equiv y \pmod n$ means $n \mid (x-y)$) is the headline example and an equivalence relation — proven via reflexivity, symmetry, and (as exercise) transitivity.
• The equivalence class $[a] = \{x : x \mathrel{R} a\}$ collects everything related to $a$; $a \mathrel{R} b \iff [a] = [b]$, so distinct classes never overlap.
• Equivalence classes always form a partition (nonempty, pairwise disjoint, union is $A$), and every partition's same-part relation is an equivalence relation — the two notions are interchangeable.
• Equivalence classes give a clean counting tool: distinct outcomes = total arrangements $\div$ size of each class, which produces the multinomial formula such as $\dfrac{11!}{4!\,4!\,2!}$ for "MISSISSIPPI."