Sets · step 1 of 13

Set-builder notation & membership

Set-builder notation describes a set by a property rather than a list: {x ∈ ℤ : x² < 10} is a set of integers, and unpacking it means testing the property element by element. Two conventions matter throughout — sets are unordered and ignore repetition, so {1, 2, 2} and {2, 1} denote the same set, and the ∈ symbol relates an element to a set while ⊆ relates two sets.

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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.

Read the notation in three parts{ variable ∈ domain : condition }. The domain limits what may be considered; the condition filters it.
Test candidates against the conditionTo list the elements, run through the domain and keep those satisfying the property. Bound the search using the condition itself.
Distinguish ∈ from ⊆1 ∈ {1, 2} is true; {1} ∈ {1, 2} is false but {1} ⊆ {1, 2} is true. Objects can also be sets, so read carefully.
Normalise before comparingOrder and duplicates carry no information. Sort and de-duplicate both sets before deciding equality.

Worked example

List the elements of A = {x ∈ ℤ : x² ≤ 9} and decide whether {0, 3} ⊆ A and whether 3 ∈ A.

The condition x² ≤ 9 means −3 ≤ x ≤ 3.
Integers in that range: −3, −2, −1, 0, 1, 2, 3.
So A = {−3, −2, −1, 0, 1, 2, 3}, with |A| = 7.
3 ∈ A since 3² = 9 ≤ 9; and both 0 and 3 lie in A, so {0, 3} ⊆ A.

Answer. A = {−3, −2, −1, 0, 1, 2, 3}; 3 ∈ A is true and {0, 3} ⊆ A is true.

Where marks get dropped

These are the specific errors that cost credit on set-builder notation & membership questions — QED's rubric penalises each of them separately.

Forgetting the negative solutions when the condition involves a square. x² ≤ 9 admits −3, not just 3.
Confusing ∈ and ⊆. Writing {1} ∈ {1, 2} is a genuine error unless the set literally contains the set {1} as an element.
Ignoring the stated domain. {x ∈ ℕ : x² ≤ 9} and {x ∈ ℤ : x² ≤ 9} have four and seven elements respectively.

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Set-builder notation & membership — frequently asked questions

Does {1, 2, 2, 3} equal {1, 2, 3}?

Yes. A set records only which objects belong, not how many times they are written or in what order. If multiplicity matters you need a multiset, which is a different structure.

What is the difference between ∅ and {∅}?

∅ is the empty set with no elements; {∅} is a set containing one element, which happens to be the empty set. So |∅| = 0 and |{∅}| = 1.

Is a colon or a vertical bar correct in set-builder notation?

Both are standard — {x : P(x)} and {x | P(x)} mean the same. Use whichever your course uses and stay consistent within an answer.

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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