Practice Equivalence Classes in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy

Which of

\frac{3}{4}

and

\frac{9}{12}

are in the same equivalence class of fractions?

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

medium

How many equivalence classes does congruence mod

n

create, and why exactly that many?

Example 4

medium

\mathbb{R}

, define

x \sim y

iff

x - y \in \mathbb{Z}

. Describe the equivalence class of

1/3

Example 5

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 6

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 7

medium

Clock positions are equivalent mod

12

. What hour is

25

o'clock equivalent to?

Example 8

hard

How many equivalence relations are there on a

3

-element set

\{a, b, c\}

Example 9

easy

Is the relation

a \le b

\mathbb{R}

an equivalence relation?

Example 10

easy

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

Example 11

easy

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

Example 12

challenge

Define

a\sim b

on integers if

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

. How many equivalence classes of residues are there?

Example 13

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 14

easy

To which class mod

4

does

13

belong?

Example 15

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 16

easy

How many equivalence classes mod

5

partition the integers?

Example 17

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 18

medium

Triangles are equivalent under congruence. Are a

3

4

5

triangle and a

5

4

3

triangle in the same class?

Example 19

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 20

easy

How many equivalence classes mod

3

partition the integers?

← Back to Equivalence Classes Worked Examples Formula Explained

Related Concepts

Set Equivalence Abstraction