Sets · step 2 of 13

Union, intersection, difference & complement

The four basic operations combine sets: A ∪ B collects everything in either, A ∩ B keeps what is in both, A \ B removes B’s elements from A, and Aᶜ is everything in the universe U outside A. The complement is the only one that depends on U, which is why every complement question must state the universe — without it, Aᶜ is undefined.

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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.

Write down the universeComplement and any expression containing it are meaningless until U is fixed.
Work innermost brackets firstSet expressions follow the same precedence discipline as arithmetic. Compute A ∩ B before uniting it with anything.
Translate each operation to a membership testx ∈ A ∪ B iff x ∈ A or x ∈ B; x ∈ A \ B iff x ∈ A and x ∉ B. Checking element by element is error-proof for small sets.
Present the answer sortedList elements in a canonical order so that comparison with the model answer is unambiguous.

Worked example

With U = {1, …, 8}, A = {1, 2, 3, 4} and B = {3, 4, 5}, compute (A ∪ B)ᶜ and A \ B.

A ∪ B = {1, 2, 3, 4, 5}.
Complement within U: everything in {1,…,8} not listed above.
(A ∪ B)ᶜ = {6, 7, 8}.
A \ B removes 3 and 4 from A, leaving {1, 2}.

Answer. (A ∪ B)ᶜ = {6, 7, 8} and A \ B = {1, 2}.

Where marks get dropped

These are the specific errors that cost credit on union, intersection, difference & complement questions — QED's rubric penalises each of them separately.

Computing a complement without a universe, or against the wrong universe. The answer changes completely with U.
Treating A \ B as symmetric. A \ B and B \ A are generally different; in the example B \ A = {5}.
Assuming A ∪ B has |A| + |B| elements. Shared elements are counted once, which is what inclusion–exclusion corrects.

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Answer in real notation

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Union, intersection, difference & complement — frequently asked questions

Is A \ B the same as A ∩ Bᶜ?

Yes, and that identity is often the cleanest way to prove statements about differences — it converts a difference into an intersection where the algebraic laws apply directly.

What is A \ A?

The empty set. Nothing is in A while also not in A. Similarly A ∪ Aᶜ = U and A ∩ Aᶜ = ∅.

Do these operations distribute?

Yes: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). Both directions hold, unlike arithmetic where only one does.

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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