A set A is a subset of B (written A ⊆ B) if every element of A is also in B. Every set is a subset of itself, and the empty set is a subset of every set.

What is the difference between subset and proper subset?

A ⊆ B means A is a subset of B (could be equal). A ⊂ B means A is a proper subset — A is contained in B but A ≠ B. Proper subsets exclude the set itself.

Is the empty set a subset of every set?

Yes. The empty set ∅ ⊆ A for any set A. Since ∅ has no elements, there's no element that could fail to belong to A. The condition is vacuously true.

If A ⊆ B, then B is a superset of A (written B ⊇ A). Superset reverses the perspective — it means B contains A. The relationship is the same, just stated differently.

How many subsets does a set with n elements have?

A set with n elements has 2ⁿ subsets. Each element can be included or excluded independently, giving 2 × 2 × ... × 2 = 2ⁿ combinations.

How many proper subsets does a set have?

A set with n elements has 2ⁿ − 1 proper subsets. This excludes the set itself from the count of all 2ⁿ subsets.

The power set P(A) is the set of all subsets of A. For A = {1, 2}, the power set is {∅, {1}, {2}, {1, 2}}. It always contains 2ⁿ elements.

What is the power set of the empty set?

P(∅) = {∅} — a set containing one element (the empty set itself). This is not empty; it has cardinality 1 = 2⁰.

Why does the empty set have one subset?

The only subset of ∅ is ∅ itself. Using the formula: 2⁰ = 1 subset. The empty set is always a subset of any set, including itself.

How does subset relate to power set?

B ⊆ A if and only if B ∈ P(A). The power set transforms subset relationships into membership — being a subset of A means being an element of A's power set.

Subsets

Summary: Subset and Superset Notation

Containment and the Power Set

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

Superset— the inverse of the subset relation

Power Set— the collection of all subsets of a given set

Empty Set— subset of every set by vacuous truth

Cardinality— determines the number of subsets via

2^{|A|}

See All Set Theory Definitions →

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 \subseteq B

The formal definition uses a universal quantifier:

A \subseteq B \iff \forall x\,(x \in A \Rightarrow x \in B)

Consider

A = \{2, 4\}

and

B = \{1, 2, 3, 4, 5\}

. Since both

2

and

4

appear in

B

, we have

A \subseteq B

. However,

\{2, 6\} \not\subseteq B

because

6 \notin B

— a single missing element breaks the subset relationship.

Two facts hold universally:

$A \subseteq A$ for any set $A$

$\emptyset \subseteq A$ 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

A

is a subset of

B

but

A

is not equal to

B

. This is written:

A \subset B

Equivalently:

A \subset B \iff (A \subseteq B) \land (A \neq B)

The distinction matters:

\{1, 2\} \subseteq \{1, 2\}

is true, but

\{1, 2\} \subset \{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 \subset B

because

c \in B

but

c \notin A

.

The empty set satisfies

\emptyset \subset A

for every non-empty set

A

, since

\emptyset \subseteq A

and

\emptyset \neq A

. However,

\emptyset \subset \emptyset

is false because

\emptyset = \emptyset

Superset

The superset relation is the inverse of the subset relation. If

A \subseteq B

, then

B

is a superset of

A

, written:

B \supseteq A

This is read as "

B

contains

A

" or "

B

is a superset of

A

." Similarly, if

A \subset B

, then

B

is a proper superset of

A

B \supset A

The superset notation simply reverses perspective. Whether to write

A \subseteq B

B \supseteq 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:

\mathbb{C} \supset \mathbb{R} \supset \mathbb{Q} \supset \mathbb{Z} \supset \mathbb{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

2^n

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:

2 \times 2 \times \cdots \times 2 = 2^n

independent choices, yielding

2^n

distinct subsets.

For proper subsets, we exclude the set itself from the count, giving

2^n - 1

proper subsets.

Examples:

$0$ elements (the empty set) has $2^0 = 1$ subset: itself

$3$ elements has $2^3 = 8$ subsets and $7$ proper subsets

$10$ elements has $2^{10} = 1024$ subsets

This counting principle connects directly to the binomial coefficients: the number of subsets of size

k

from a set of

n

elements is

\binom{n}{k}

, and summing over all possible sizes gives:

\sum_{k=0}^{n} \binom{n}{k} = 2^n

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

\mathcal{P}(A)

2^A

, is the set of all subsets of

A

\mathcal{P}(A) = \{S : S \subseteq A\}

For

A = \{1, 2\}

, the power set is:

\mathcal{P}(\{1, 2\}) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}

The empty set and

A

itself are always elements of

\mathcal{P}(A)

.

The cardinality of the power set follows directly from subset counting:

|\mathcal{P}(A)| = 2^{|A|}

The power set of the empty set deserves attention:

\mathcal{P}(\emptyset) = \{\emptyset\}

This is not the empty set — it is a set containing one element (the empty set). The distinction matters:

|\mathcal{P}(\emptyset)| = 1

, not

0

.

The power set transforms subset relationships into membership relationships:

B \subseteq A

if and only if

B \in \mathcal{P}(A)

Summary: Subset and Superset Notation

Four containment symbols cover every subset/superset relationship encountered above. The capstone reference card below collects all four — with how each is read, what it means, and a representative example — so the right notation is always one glance away.

Symbol	Read as	Meaning	Example
A ⊆ B	"A is a subset of B"	every element of A is in B (A may equal B)	{1, 2} ⊆ {1, 2}; ∅ ⊆ A for any A
A ⊂ B	"A is a proper subset of B"	A ⊆ B and A ≠ B	{1, 2} ⊂ {1, 2, 3}
B ⊇ A	"B is a superset of A"	inverse of ⊆; B contains every element of A	ℝ ⊇ ℚ
B ⊃ A	"B is a proper superset of A"	inverse of ⊂; B properly contains A	ℝ ⊃ ℚ; ℂ ⊃ ℝ ⊃ ℚ ⊃ ℤ ⊃ ℕ