Subsets & the power set π«(A)
A β B means every element of A is in B; the power set π«(A) is the set of all subsets of A, and it always has 2βΏ elements for |A| = n. Two boundary cases catch people out constantly: β is a subset of every set, and A is a subset of itself, so both always appear in π«(A).
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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.
- Count firstFor |A| = n, expect exactly 2βΏ subsets. Knowing the target count tells you immediately whether your list is complete.
- List by sizeWrite the empty set, then all singletons, then all pairs, and so on up to A itself. This ordering makes omissions visible.
- Check β element by elementTo verify A β B, confirm each element of A lies in B. To refute it, one element of A outside B suffices.
- Keep β and β apartElements of π«(A) are SETS. So {1} β π«(A) and {1} β A are both correct, while 1 β π«(A) is not.
Worked example
List π«({a, b, c}) and state |π«({a, b, c})|.
- |A| = 3, so there are 2Β³ = 8 subsets.
- Size 0: β . Size 1: {a}, {b}, {c}.
- Size 2: {a, b}, {a, c}, {b, c}. Size 3: {a, b, c}.
- Count the list: 1 + 3 + 3 + 1 = 8 β.
Answer. π«({a,b,c}) = {β , {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}, of cardinality 8.
Where marks get dropped
These are the specific errors that cost credit on subsets & the power set π«(a) questions β QED's rubric penalises each of them separately.
- Omitting β or A itself from the power set. They are always members, and the 2βΏ count exposes the omission.
- Writing elements instead of sets inside π«(A) β listing a rather than {a}.
- Confusing β with β. Many courses use β for proper subset, excluding A itself; check your convention before answering.
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Subsets & the power set π«(A) β frequently asked questions
Why does |π«(A)| = 2βΏ?
Building a subset means making an independent in/out choice for each of the n elements, so there are 2 Γ 2 Γ β¦ Γ 2 = 2βΏ subsets. This bijection with binary strings is the standard proof.
What is π«(β )?
{β } β a set with exactly one element. And π«({β }) = {β , {β }}, with two. The 2βΏ formula holds: 2β° = 1.
How many subsets of size k does an n-set have?
C(n, k), the binomial coefficient. Summing over all k gives Ξ£ C(n,k) = 2βΏ, which is another proof of the power-set count.
The rest of Sets
Set-builder notation, operations, and set identities. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Set-builder notation & membership
- 2Union, intersection, difference & complement
- 3Subsets & the power set π«(A)
- 4The Cartesian product A Γ B
- 5Proving set identities
- 6Cardinality & inclusionβexclusion
- 7Indexed families & generalised β / β
- 8Partitions & disjoint unions
- 9Countable vs uncountable sets
- 10Characteristic (indicator) functions
- 11Venn diagrams & shading regions
- 12Symmetric difference A β³ B
- 13Russellβs paradox & naive set theory
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