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.
Key Terms
Subset— a set whose elements all belong to another set
Proper Subset— a subset strictly smaller than its containing set
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.
UAB
A ⊆ B, frozen
The whole of circle A sits inside circle B — shaded, because every element of A is already an element of B. Nudge the sets between nested, overlapping, and disjoint configurations on the two-set Venn explorer.
Containment is the relation the rest of this page builds on — proper subsets sharpen it, supersets reverse it, and the power set collects every way it can happen.
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:
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
n = |A|
Total subsets (2n)
Proper subsets (2n − 1)
0
1
0
1
2
1
2
4
3
3
8
7
4
16
15
5
32
31
10
1,024
1,023
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:
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).
Subset and Superset Notation
Notation
Subset and Superset Notation
Four containment marks, a calligraphic P with an exponent twin, and the double-struck number chain — all met above, gathered here with their reading rules and their one genuine convention war. The reference card below lists the four containment symbols at a glance.
Containment with and without the escape clause: A⊆B allows A=B, A⊂B forbids it — the underbar means "or equal", the same bar-job it performs on the inequality signs. Subsets and Proper Subsets above define both.
CasesThe convention war: many texts — especially older and European ones — write ⊂ for any subset and reserve ⊊ for proper containment; when it matters, an author's first ⊂ deserves a suspicious read.
Do not confuseMembership. 1∈A but {1}⊆A — the basics page calls this the subject's most reliable exam error, and it runs both directions.
⊇ · ⊃
is a superset of; contains
The same relationships read from the other end: B⊇A says B contains A — one fact, two spellings, mirror-image marks. Superset above works the reversal.
CasesThe open side faces the bigger set — the same visual rule inequality signs follow; A⊆B and B⊇A are interchangeable, and writers pick whichever keeps the emphasized set first.
Do not confuseA different claim. Reversing the glyph does not negate it — the denial of A⊆B is A⊆B with the slash, not A⊇B, which asserts containment the other way.
P(A) · 2A
the power set of A; two to the A
The set of all subsets, in calligraphic dress — P(A)={S:S⊆A}, per Power Set above. The exponent spelling 2A earns its name from the count: ∣P(A)∣=2∣A∣.
CasesThe bridge identity converts relations: B⊆A exactly when B∈P(A) — containment outside becomes membership inside; P(∅)={∅}, one element, not zero.
Do not confuseA plain capital P. The calligraphic font is load-bearing — P(A) in roman type reads as probability of an event or a polynomial value; the script P keeps the power set visually distinct.
N⊂Z⊂Q⊂R
the number systems, each inside the next
The double-struck (blackboard-bold) family: naturals, integers, rationals, reals — reserved letters for the standard number sets, and the containment chain is the family portrait. Chalk invented the double stroke to fake print bold; print then adopted it back.
CasesThe chain extends to C at the top — the complex numbers own that link — and each letter serves as a standing domain, n∈Z, x∈R, wherever a variable's habitat needs declaring.
Do not confuseOrdinary italic letters. N, Z, Q, R are free variables; only the double stroke reserves the letter — handwriting that skips the second stroke silently releases the reservation.
What is the difference between a subset and a proper subset?
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A subset allows equality, a proper subset does not. A ⊆ B holds when every element of A is in B, including the case A = B. A ⊂ B additionally requires at least one element of B to be missing from A. So {1, 2} ⊆ {1, 2} is true, but {1, 2} ⊂ {1, 2} is false.Read more →
Is the empty set a subset of every set?
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Yes. The subset rule asks whether every element of the empty set also belongs to the other set. Since the empty set has no elements, there is nothing that could fail the test, so the condition holds automatically. This is called vacuous truth. It follows that ∅ ⊆ A for every set A, including ∅ itself.Read more →
How many proper subsets does a set have?
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A set with n elements has 2ⁿ − 1 proper subsets. The full count of subsets is 2ⁿ, since each element is either included or excluded, and proper subsets exclude only one case: the set itself. A set of 3 elements therefore has 8 subsets and 7 proper subsets. The subset counting table lists more values.Read more →
What is the power set of the empty set?
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It is {∅}, a set containing one element. The power set collects every subset, and the empty set has exactly one subset — itself. So 𝒫(∅) is not empty: its cardinality is 1, not 0. Confusing {∅} with ∅ is the standard error here; one is a box holding an empty box, the other is no box at all.Read more →
What is the difference between ∈ and ⊆?
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The first is membership, the second is containment. Write 1 ∈ A when 1 is an element of A, and {1} ⊆ A when the whole set {1} sits inside A. The left side of ∈ is an element; the left side of ⊆ is a set. Mixing them is the most common error in set theory, and it runs in both directions.Read more →
Does ⊂ always mean proper subset?
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No. This site and most modern texts use ⊂ for proper containment and ⊆ when equality is allowed. But many older and European authors write ⊂ for any subset and reserve ⊊ for the proper case. The symbol alone is not enough — check how a given text defines it before trusting the distinction. See all set theory symbols.Read more →