Equivalence Relations · step 3 of 13

The class–partition correspondence

Equivalence relations and partitions are two views of the same information. Given an equivalence relation, its classes partition the set; given a partition, "lies in the same block" is an equivalence relation. The two constructions are mutually inverse, which is why counting equivalence relations on an n-set is the same as counting partitions of it.

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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.

Relation → partitionCompute the classes. They are automatically non-empty, disjoint and covering — that is the fundamental theorem.
Partition → relationDefine a ∼ b iff a and b lie in the same block. Reflexivity, symmetry and transitivity follow from the block structure.
Check the round tripStarting from a relation, forming the partition and going back gives the original relation. The correspondence is a bijection.
Count via partitionsSince the correspondence is a bijection, the number of equivalence relations on an n-set is the Bell number Bₙ.

Worked example

Write down the equivalence relation on {1,2,3} induced by the partition {{1,3},{2}}, and count its pairs.

Same block pairs from {1,3}: (1,1), (1,3), (3,1), (3,3).
Same block pairs from {2}: (2,2).
So ∼ = {(1,1),(1,3),(3,1),(3,3),(2,2)}, with 5 pairs.
Check: reflexive (all three loops present), symmetric (1,3 and 3,1), transitive ✓.

Answer. ∼ = {(1,1),(2,2),(3,3),(1,3),(3,1)} — 5 ordered pairs, and the classes recover the original partition.

Where marks get dropped

These are the specific errors that cost credit on the class–partition correspondence questions — QED's rubric penalises each of them separately.

Omitting the loops when converting a partition to a relation. Every element is in the same block as itself, so all pairs (a,a) belong.
Allowing an empty block. A partition’s blocks must be non-empty, or the correspondence with equivalence relations breaks.
Assuming a union of two equivalence relations is one. It usually is not — the corresponding partitions do not combine that way.

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The class–partition correspondence — frequently asked questions

How many equivalence relations are on a 4-element set?

B₄ = 15, matching the 15 partitions of a 4-set. Listing them by block-size pattern (4; 3+1; 2+2; 2+1+1; 1+1+1+1) gives 1 + 4 + 3 + 6 + 1 = 15.

Is the correspondence really a bijection?

Yes. Classes-of-a-relation and same-block-relation are inverse constructions, and proving both round trips is a standard exam question.

What partition corresponds to equality?

The finest one — all singletons. The coarsest partition, a single block, corresponds to the total relation where everything is equivalent to everything.

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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