Equivalence Classes Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Equivalence Classes.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An equivalence class is the set of all elements that are related to a given element under an equivalence relation — it groups objects that are considered 'the same' in some specified sense.

Treating different things as equal because they share what matters.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An equivalence class collects every element related to a given one, treating them all as a single object.

Common stuck point: The procedure for equivalence classes is the easy part; the trap is grouping by a relation that is not reflexive, symmetric, and transitive. Asking "Am I grouping all elements 'the same' under a reflexive, symmetric, and transitive relation?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I grouping all elements 'the same' under a reflexive, symmetric, and transitive relation?

Worked Examples

Example 1

medium

Define the relation

a \sim b

\mathbb{Z}

by '

a \sim b

iff

a \equiv b \pmod{3}

'. Verify this is an equivalence relation and list the equivalence classes.

Answer

\mathbb{Z}/3\mathbb{Z} = \{[0],[1],[2]\}

First step

Reflexive:

a \equiv a \pmod{3}

(since

3 \mid 0

). True.

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Example 2

medium

On the set of triangles, define

T_1 \sim T_2

T_1

is congruent to

T_2

. Show this is an equivalence relation.

Example 3

medium

Define

a \sim b

\mathbb{Z}

a \sim b

iff

5 \mid (a - b)

. List the equivalence class

[7]

in roster form (first few positive and negative members).

Example 4

medium

Show that 'same first letter' on the set of English words is an equivalence relation, and count the classes.

Example 5

medium

\mathbb{R}

, define

x \sim y

iff

x - y \in \mathbb{Z}

. Describe the equivalence class of

1/3

Example 6

hard

Show that congruence of triangles is an equivalence relation, and explain why the set of all triangles modulo congruence is a useful quotient.

Example 7

hard

Prove that if

\sim

is an equivalence relation on

S

, then any two equivalence classes are either equal or disjoint.

Example 8

hard

Verify whether

a \sim b

defined on

\mathbb{Z}

a \sim b

iff

a + b

is even is an equivalence relation. If yes, describe the classes.

Example 9

challenge

How many equivalence relations are there on a

4

-element set

\{a, b, c, d\}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

List the equivalence classes of

\mathbb{Z}

under

a \sim b

iff

a \equiv b \pmod{2}

Example 2

medium

Define

f \sim g

on functions from

\mathbb{R}

\mathbb{R}

f \sim g

iff

f(0) = g(0)

. Verify this is an equivalence relation and describe the equivalence class of

f(x) = x^2

Example 3

easy

How many equivalence classes mod

3

partition the integers?

Example 4

easy

Are

\frac{1}{2}

and

\frac{2}{4}

in the same equivalence class of fractions?

Example 5

easy

To which class mod

4

does

13

belong?

Example 6

easy

How many equivalence classes does the relation 'same parity' create on the integers?

Example 7

easy

Is 'has the same number of letters' an equivalence relation on words?

Example 8

easy

How many equivalence classes mod

2

are there, and name them.

Example 9

easy

Is the relation

a < b

an equivalence relation on integers?

Example 10

easy

Which fraction is the simplest representative of the class of

\frac{6}{9}

Example 11

medium

Under congruence mod

5

, what single class does

7+9

belong to?

Example 12

medium

How many equivalence classes does congruence mod

n

create, and why exactly that many?

Example 13

medium

Vectors are 'equivalent' if they have the same direction and length (free vectors). Are

\langle1,2\rangle

based at the origin and

\langle1,2\rangle

based at

(3,3)

in the same class?

Example 14

medium

Triangles are equivalent under congruence. Are a

3

4

5

triangle and a

5

4

3

triangle in the same class?

Example 15

medium

On the rationals, define

a\sim b

a-b

is an integer. Which class contains

\frac{7}{2}

Example 16

medium

Is 'is a friend of' (assume non-reflexive, can be one-directional) an equivalence relation? Identify which property may fail.

Example 17

challenge

Under rotation, two colorings of a bracelet of

3

beads (each black or white) are equivalent. How many equivalence classes are there?

Example 18

challenge

Define

a\sim b

on integers if

a^2\equiv b^2\pmod{8}

. How many equivalence classes of residues are there?

Example 19

challenge

Slopes define equivalence: two nonvertical lines are equivalent if parallel (equal slope). Lines

y=2x+1

y=2x-3

y=3x

give how many classes among these three?

Example 20

medium

Under congruence mod

7

, which class does

100

belong to?

Example 21

medium

Clock positions are equivalent mod

12

. What hour is

25

o'clock equivalent to?

Example 22

medium

How many equivalence classes does 'congruent triangles' create among all triangles with sides from

\{3,4,5\}

used once each?

Example 23

easy

How many equivalence classes mod

5

partition the integers?

Example 24

easy

To which class mod

7

does the integer

30

belong?

Example 25

easy

Is the relation 'has the same birthday' on people an equivalence relation?

Example 26

easy

Is the relation

a \le b

\mathbb{R}

an equivalence relation?

Example 27

easy

Which of

\frac{3}{4}

and

\frac{9}{12}

are in the same equivalence class of fractions?

Example 28

medium

On the set of nonzero integers, define

a \sim b

iff

ab > 0

. How many equivalence classes are there?

Example 29

medium

\mathbb{R}^2

, define

(x_1, y_1) \sim (x_2, y_2)

iff

x_1^2 + y_1^2 = x_2^2 + y_2^2

. Describe the equivalence class containing

(3, 4)

Example 30

medium

How many distinct equivalence classes of integers mod

12

does the relation '

a \equiv b \pmod{12}

' produce?

Example 31

medium

Is the relation 'is a sibling of' (with the convention that a person is their own sibling) an equivalence relation on a set of children sharing parents?

Example 32

medium

On the set of integers, define

a \sim b

iff

a

and

b

have the same number of digits (in base 10). Is this an equivalence relation on positive integers?

Example 33

medium

How many equivalence classes does '

a \equiv b \pmod{8}

' have on the set

\{0, 1, 2, \dots, 23\}

Example 34

hard

\mathbb{Z}

, define

a \sim b

iff

a^2 \equiv b^2 \pmod 5

. How many equivalence classes does this produce on

\{0, 1, 2, 3, 4\}

Example 35

hard

How many equivalence relations are there on a

3

-element set

\{a, b, c\}

Example 36

hard

On the set of

2 \times 2

real matrices, define

A \sim B

iff

\det A = \det B

. Describe the equivalence class of the identity matrix

I

Example 37

challenge

Let

\sim

\mathbb{Z}^2

be defined by

(a, b) \sim (c, d)

iff

a + d = b + c

. Describe the equivalence class of

(0, 0)

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Related Concepts

Set Equivalence Abstraction

Background Knowledge

These ideas may be useful before you work through the harder examples.

setequivalence