Equivalence Relations · step 2 of 13

Equivalence classes [a]

The class of a is [a] = {x : x ∼ a}, the set of everything equivalent to it. The central theorem is that two classes are either identical or disjoint — never partially overlapping — and [a] = [b] exactly when a ∼ b. That is why classes can be named by any of their members, and why choosing a representative is always legitimate.

Practise equivalence classes [a] free → All Equivalence Relations subtopics

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Compute one class at a timePick an element, collect everything related to it, and record the class. Cross those elements off.
Continue with an uncovered elementRepeat with anything not yet in a class. The process terminates when everything is covered.
Use the equality criterionTo decide whether [a] = [b], test a ∼ b. You never need to compare the sets element by element.
Count and checkThe classes must be non-empty, pairwise disjoint, and cover the whole set — their sizes must sum to the size of the set.

Worked example

For congruence mod 4 on ℤ, describe all equivalence classes and decide whether [7] = [15].

[0] = {…, −4, 0, 4, 8, …}, the multiples of 4.
[1], [2], [3] are the integers leaving remainder 1, 2, 3 respectively.
Every integer has exactly one remainder in {0,1,2,3}, so there are exactly four classes.
For [7] and [15]: 15 − 7 = 8, and 4 ∣ 8, so 7 ∼ 15.

Answer. Four classes [0], [1], [2], [3]; and [7] = [15] = [3], since both leave remainder 3.

Where marks get dropped

These are the specific errors that cost credit on equivalence classes [a] questions — QED's rubric penalises each of them separately.

Thinking [7] and [3] are different classes because the representatives differ. Representatives are not canonical — [7] = [3] mod 4.
Allowing classes to overlap. If you compute two classes sharing an element, you have made an arithmetic error: they must be equal.
Forgetting that a ∈ [a] always. Reflexivity guarantees every class is non-empty and contains its own representative.

Practise this until it is automatic

Unlimited fresh questions

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Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.

Saved to your library

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Equivalence classes [a] — frequently asked questions

Why are classes either equal or disjoint?

If x ∈ [a] ∩ [b] then a ∼ x and x ∼ b, so a ∼ b by transitivity and symmetry, hence [a] = [b]. Any overlap forces full equality.

How do I choose a canonical representative?

Pick a systematic rule — for mod n, the remainder in {0,…,n−1}. Canonical representatives make answers comparable but are a convention, not a mathematical requirement.

Can classes have different sizes?

Yes. For "same absolute value" on ℤ, the class of 0 is a singleton while every other class has two elements. Only special relations give uniformly sized classes.

The rest of Equivalence Relations

Equivalence classes, partitions and quotient sets. Each subtopic below has its own method, worked example and mark-losing traps.

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