Equivalence Relations · step 4 of 13

The quotient set A/∼

The quotient set A/∼ is the set whose elements are the equivalence classes of ∼. It formalises "identifying" equivalent things: ℤ/≡₅ has five elements even though ℤ is infinite, because everything congruent mod 5 has been collapsed to one point. The canonical projection π : A → A/∼ sending a to [a] is always surjective, and its kernel is exactly ∼.

Practise the quotient set 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 the classes firstThe elements of A/∼ ARE the classes, so you cannot describe the quotient before you know them.
Count the classes|A/∼| is the number of classes, not the number of elements of A. For a finite A the class sizes sum to |A|.
Describe the projectionπ(a) = [a] maps each element to its class. It is surjective by construction and injective only when ∼ is equality.
Name the quotient if it is familiarℤ/nℤ, the set of directions of lines, or the set of remainders — recognising the quotient is usually the point of the question.

Worked example

On A = {1,…,10}, let a ∼ b iff a and b have the same remainder on division by 3. Describe A/∼ and its class sizes.

Remainder 0: {3, 6, 9} — three elements.
Remainder 1: {1, 4, 7, 10} — four elements.
Remainder 2: {2, 5, 8} — three elements.
Sizes sum to 3 + 4 + 3 = 10 = |A| ✓.

Answer. A/∼ = {[3], [1], [2]} has three elements — the classes of sizes 3, 4 and 3.

Where marks get dropped

These are the specific errors that cost credit on the quotient set a/∼ questions — QED's rubric penalises each of them separately.

Writing elements of A as elements of A/∼. The quotient contains sets, so its elements are [1] and [2], not 1 and 2.
Assuming |A/∼| divides |A|. Only when all classes have the same size — the example above has classes of sizes 3, 4, 3.
Treating the projection as injective. It is injective exactly when every class is a singleton, i.e. when ∼ is equality.

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Answer in real notation

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The quotient set A/∼ — frequently asked questions

Why bother with quotients?

They build new objects out of old ones: ℤ/nℤ gives modular arithmetic, quotient groups give factor structures, and quotienting a set of states by "behaves the same" is how automata are minimised.

Is A/∼ a subset of A?

No — it is a subset of 𝒫(A), since its elements are subsets of A. Confusing the two levels is the most common error in quotient questions.

What is the universal property?

Any function f on A that is constant on classes factors uniquely through π, giving a well-defined function on A/∼. That factorisation is precisely what "well-defined on classes" means.

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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