Sets · step 8 of 13

Partitions & disjoint unions

A partition of A is a collection of non-empty, pairwise disjoint subsets whose union is A. Every element lands in exactly one block — that "exactly one" is the whole content of the definition, and it is what makes partitions equivalent to equivalence relations. The counting payoff is the addition rule: for a partition into blocks, |A| is the sum of the block sizes.

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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.

Check all three conditionsEach block non-empty, blocks pairwise disjoint, union equals A. A proposed partition failing any one is not a partition.
Test disjointness pairwiseWith k blocks there are C(k,2) pairs to check; a single shared element disqualifies the collection.
Confirm the union covers AEvery element of A must appear somewhere. Missing elements are the most common failure.
Count with the addition ruleFor a genuine partition, |A| = Σ |Bᵢ|. If the sum overshoots, the blocks were not disjoint.

Worked example

Is {{1,2}, {3}, {4,5}} a partition of A = {1,2,3,4,5}? What about {{1,2},{2,3},{4,5}}?

First collection: all three blocks are non-empty ✓.
Pairwise intersections are all empty ✓; the union is {1,2,3,4,5} = A ✓. Sizes 2+1+2 = 5 = |A| ✓.
Second collection: {1,2} ∩ {2,3} = {2} ≠ ∅.
Disjointness fails, and the size sum 2+2+2 = 6 exceeds |A| = 5.

Answer. The first is a partition into 3 blocks; the second is not, because 2 lies in two blocks.

Where marks get dropped

These are the specific errors that cost credit on partitions & disjoint unions questions — QED's rubric penalises each of them separately.

Allowing the empty set as a block. Blocks must be non-empty, or the number of blocks becomes meaningless.
Checking only that the union is A. Overlapping blocks can still cover A without partitioning it.
Confusing a partition of a set with a partition of an integer — the second is a way of writing n as a sum, an entirely different object.

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Partitions & disjoint unions — frequently asked questions

How many partitions does an n-element set have?

The Bell number Bₙ: 1, 1, 2, 5, 15, 52, 203 for n = 0…6. There is no simple closed form, but the recurrence Bₙ₊₁ = Σ C(n,k) Bₖ generates them.

What is the link to equivalence relations?

They are the same information. Every equivalence relation partitions its set into equivalence classes, and every partition defines an equivalence relation by "in the same block".

Can a partition have infinitely many blocks?

Yes. Partitioning ℤ by residue mod n gives n blocks, but partitioning ℝ into singletons gives uncountably many. The definition puts no bound on the number.

The rest of Sets

Set-builder notation, operations, and set identities. Each subtopic below has its own method, worked example and mark-losing traps.

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