A subset relationship describes when one set is entirely contained within another. This concept provides the fundamental way to compare sets and establish hierarchies among collections. From subsets emerges the power set — the collection of all possible subsets a given set can produce — which connects directly to combinatorics and counting principles.
Subsets
A set A is a subset of a set B if every element of A is also an element of B. This relationship is written:
A⊆B
The formal definition uses a universal quantifier:
A⊆B⟺∀x(x∈A⇒x∈B)
Consider A={2,4} and B={1,2,3,4,5}. Since both 2 and 4 appear in B, we have A⊆B. However, {2,6}⊆B because 6∈/B — a single missing element breaks the subset relationship.
Two facts hold universally:
A⊆A for any set A
∅⊆A for any set A
The first fact is immediate since every element of A trivially belongs to A. The second follows because the empty set contains no elements, so there is no element that could fail to belong to A.
Proper Subsets
A set A is a proper subset of B if A is a subset of B but A is not equal to B. This is written:
A⊂B
Equivalently:
A⊂B⟺(A⊆B)∧(A=B)
The distinction matters: {1,2}⊆{1,2} is true, but {1,2}⊂{1,2} is false because the two sets are equal. For a proper subset, there must exist at least one element in B that does not belong to A.
If A={a,b} and B={a,b,c}, then A⊂B because c∈B but c∈/A.
The empty set satisfies ∅⊂A for every non-empty set A, since ∅⊆A and ∅=A. However, ∅⊂∅ is false because ∅=∅.
Superset
The superset relation is the inverse of the subset relation. If A⊆B, then B is a superset of A, written:
B⊇A
This is read as "B contains A" or "B is a superset of A." Similarly, if A⊂B, then B is a proper superset of A:
B⊃A
The superset notation simply reverses perspective. Whether to write A⊆B or B⊇A depends on which set you want to emphasize. Both statements convey the same relationship.
The standard number sets provide a natural example of superset chains:
C⊃R⊃Q⊃Z⊃N
Each set properly contains the one to its right: the complex numbers contain the reals, which contain the rationals, and so on.
Number of Subsets
A set with n elements has exactly 2n subsets. This count includes both the empty set and the set itself.
The reasoning is straightforward: for each element, there are two choices — include it in the subset or exclude it. With n elements, this gives:
For proper subsets, we exclude the set itself from the count, giving 2n−1 proper subsets.
Examples:
0 elements (the empty set) has 20=1 subset: itself
3 elements has 23=8 subsets and 7 proper subsets
10 elements has 210=1024 subsets
This counting principle connects directly to the binomial coefficients: the number of subsets of size k from a set of n elements is (kn), and summing over all possible sizes gives:
k=0∑n(kn)=2n
Power Set
The power set of a set A, denoted P(A) or 2A, is the set of all subsets of A:
P(A)={S:S⊆A}
For A={1,2}, the power set is:
P({1,2})={∅,{1},{2},{1,2}}
The empty set and A itself are always elements of P(A).
The cardinality of the power set follows directly from subset counting:
∣P(A)∣=2∣A∣
The power set of the empty set deserves attention:
P(∅)={∅}
This is not the empty set — it is a set containing one element (the empty set). The distinction matters: ∣P(∅)∣=1, not 0.
The power set transforms subset relationships into membership relationships: B⊆A if and only if B∈P(A).