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

Groups of objects that are considered 'the same' under some equivalence relation.

Treating different things as equal because they share what matters.

Read the full concept explanation →

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: Equivalence classes partition objects into 'sameness' groups.

Common stuck point: The equivalence relation defines what 'same' means in that context.

Sense of Study hint: Pick one element and find all others that are related to it. That collection is its equivalence class. Then verify the relation is reflexive, symmetric, and transitive.

Worked Examples

Example 1

medium
Define the relation a \sim b on \mathbb{Z} by 'a \sim b iff a \equiv b \pmod{3}'. Verify this is an equivalence relation and list the equivalence classes.

Solution

  1. 1
    Reflexive: a \equiv a \pmod{3} (since 3 \mid 0). True.
  2. 2
    Symmetric: if 3 \mid (a-b), then 3 \mid (b-a). True.
  3. 3
    Transitive: if 3 \mid (a-b) and 3 \mid (b-c), then 3 \mid (a-c) (by addition). True.
  4. 4
    Equivalence classes: [0]=\{\ldots,-3,0,3,6,\ldots\}, [1]=\{\ldots,-2,1,4,7,\ldots\}, [2]=\{\ldots,-1,2,5,8,\ldots\}. These three classes partition \mathbb{Z}.

Answer

\mathbb{Z}/3\mathbb{Z} = \{[0],[1],[2]\}
An equivalence relation partitions a set into disjoint equivalence classes. Modular arithmetic provides the canonical example: every integer belongs to exactly one class [0],[1],\ldots,[n-1] modulo n.

Example 2

medium
On the set of triangles, define T_1 \sim T_2 if T_1 is congruent to T_2. Show this is an equivalence relation.

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} to \mathbb{R} by f \sim g iff f(0) = g(0). Verify this is an equivalence relation and describe the equivalence class of f(x) = x^2.
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Related Concepts

SetEquivalenceAbstraction

Background Knowledge

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

setequivalence