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Subset (mathematics)

Explanation of subsets: definitions, notation, examples, properties and distinctions such as proper subset and relation to power sets, aimed at a general mathematical audience.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: June 3, 2026

en.wikipedia.org · Public domain

A subset is a fundamental concept in set theory and everyday mathematics. Informally, a set A is called a subset of a set B when every element of A also belongs to B. This relationship is written A ⊆ B. When A contains some but not all elements of B, A is a proper (or strict) subset of B and is often written A ⊂ B. The empty set Ø is a subset of every set, while every set is a subset of itself.

Definitions and notation

Key terms and symbols used with subsets include:

Subset (non-strict): A ⊆ B means ∀x(x ∈ A ⇒ x ∈ B).
Proper subset (strict): A ⊂ B means A ⊆ B and A ≠ B.
Not a subset: A ⊄ B indicates there exists an element of A not in B.
Superset: B ⊇ A is equivalent to A ⊆ B; B ⊃ A corresponds to A ⊂ B.

Examples

Concrete examples make the idea clear. If B = {1, 2, 3}, then A = {1, 2} satisfies A ⊂ B, while C = {1, 2, 3} satisfies C ⊆ B but not C ⊂ B. The set Ø is a subset of B and of every other set. Subsets can be finite or infinite; for instance, the set of even integers is a subset of the integers.

The subset relation is reflexive (A ⊆ A for every A) and transitive (if A ⊆ B and B ⊆ C then A ⊆ C), making it a partial order on the collection of all subsets of a given set. The collection of all subsets of a set S is called the power set of S and is denoted P(S); its elements are precisely the subsets of S. Cardinality interacts with inclusion: if A ⊂ B then |A| ≤ |B|, with strict inequality when both are finite and A ≠ B.

Uses, distinctions and remarks

Subsets are used throughout mathematics to form substructures (subgroups, subspaces, suborders), to specify domains and constraints, and to reason about containment. Be aware that notation varies: some authors use ⊂ to mean non-strict subset (same as ⊆), while others reserve ⊂ for strict inclusion. When precision matters, the words "proper" or explicit equality checks help avoid ambiguity. For further reading on foundational ideas see general resources on sets.

Definition

If and are sets and each element of also an element of , then called a subset or subset of :

$A \subseteq B :\Longleftrightarrow \forall x \in A\colon x \in B$

Conversely, called the superset of exactly when is subset of

$B \supseteq A :\Longleftrightarrow A \subseteq B$

Furthermore, there is the notion of a real subset. is a real subset of exactly if a subset of and is not identical to }.

$A\subsetneq B:\Longleftrightarrow A\subseteq B\land A\neq B$

Again, one also writes $B \supsetneq A$ if $A \subsetneq B$ .

Other notations

⊂⊊⊆⊇⊋⊃

Some authors also use the characters $\subset$ and $\supset$ for subset and superset instead of $\subseteq$ and $\supseteq$ . Mostly the author then does not define the term "real subset".

Other authors prefer the characters $\subset$ and $\supset$ for true subset and superset thus instead of $\subsetneq$ and $\supsetneq$ . This usage is fittingly reminiscent of the characters for inequality ≤ $\leq$ and . Since this notation is mostly used when the difference between real and non-real subsets is important, the characters $\subsetneq$ and $\supsetneq$ rather rarely used.

Variants of the character $\subsetneq$ are also $\varsubsetneq$ , $\subsetneqq$ and $\varsubsetneqq$ . If is not a subset of , can also be $A\nsubseteq B :\Longleftrightarrow \lnot \left(A\subseteq B\right)$ used. Corresponding notations are $\varsupsetneq$ for $\supsetneq$ , $\supsetneqq$ and $\varsupsetneqq$ for $\supsetneq$ , and $A\nsupseteq B$ (no superset).

The corresponding Unicode symbols are: ⊂, ⊃, ⊆, ⊇, ⊄, ⊅, ⊈, ⊉, ⊊, ⊋ (see: Unicode block Mathematical Operators).

Speech

Instead of " is a subset of ." we also say "The set contained in the set " or "The set is contained by ." said. Similarly, instead of saying " is a superset of .", we say "The set contains the set ." or "The set contains the set .". If there can be no misunderstanding, " contains ." etc. is also said. Misunderstandings can arise in particular with "The set contains the element .".

Examples

{1, 2} is a (real) subset of {1, 2, 3}.
{1, 2, 3} is a (fake) subset of {1, 2, 3}.
{1, 2, 3, 4} is not a subset of {1, 2, 3}.
{1, 2, 3} is not a subset of {2, 3, 4}.
{} is a (real) subset of {1, 2}.
{1, 2, 3} is a (real) superset of {1, 2}.
{1, 2} is a (fake) superset of {1, 2}.
{1} is not a superset of {1, 2}.
The set of prime numbers is a real subset of the set of natural numbers.
The set of rational numbers is a real subset of the set of real numbers.

More examples as set diagrams:

A is a real subset of B

C is a subset of B, but not a real subset of B

The set {drum, playing card} is a subset of the set {guitar, playing card, digital camera, drum}

The regular polygons form a subset of the set of all polygons.

Properties

The empty set is a subset of each set:

$\varnothing \subseteq A$

Every set is a subset of itself:

$A \subseteq A$

Characterization of inclusion with the help of the association:

$A \subseteq B \Leftrightarrow A \cup B = B$

Characterization of inclusion using the average:

$A \subseteq B \Leftrightarrow A \cap B = A$

Characterization of inclusion using the difference set:

$A \subseteq B \Leftrightarrow A \setminus B = \varnothing$

Characterization of inclusion using the characteristic function:

$A \subseteq B \Leftrightarrow \chi_A \le \chi_B$

Two sets are equal if and only if each is a subset of the other:

$A=B\Leftrightarrow A\subseteq B\land B\subseteq A$

This rule is often used when proving equality of two sets by showing mutual inclusion (in two steps).

In the transition to complement, the direction of inclusion reverses:

$A \subseteq B \Rightarrow A^{\rm c} \supseteq B^{\rm c}$

When forming the intersection, you always get a subset:

$A \cap B \subseteq A$

When forming the union set, you always get a superset:

$A \cup B \supseteq A$

Inclusion as order relation

Inclusion as a relation between sets satisfies the three properties of a partial order relation, namely it is reflexive, antisymmetric and transitive:

$A \subseteq A$

$A \subseteq B \subseteq A \Rightarrow A = B$

$A \subseteq B \subseteq C \Rightarrow A \subseteq C$

(Where $A\subseteq B\subseteq C$ a shorthand notation for $A\subseteq B$ and $B\subseteq C$ .)

Thus, if $M \,$ a set of sets (a set system), then $(M, \subseteq)$ a half-order. In particular, this holds for the power set $\mathcal P(X)$ a given set .

Inclusion Chains

If is $M \,$ a set system such that of every two sets occurring in $M \,$ one includes or is included by the other, such a set system is called an inclusion chain. An example of this is provided by the system $\{{]{-\infty, x}[} \mid x \in \R \}$ of the left unconstrained open intervals of $\mathbb {R}$ .

A special case of an inclusion chain exists if a (finite or infinite) set sequence is given which is ordered by $\subseteq$ ascending or by $\supseteq$ descending. One writes then briefly:

$A_1 \subseteq A_2 \subseteq A_3 \subseteq \ ...$

$A_1 \supseteq A_2 \supseteq A_3 \supseteq \ ...$

Size and number of subsets

Every subset of a finite set is finite and for the powers holds:

$A \subseteq B \Rightarrow \left| A\right| \le \left| B\right|$

$A \subsetneq B \Rightarrow \left| A\right| < \left| B\right|$

Every superset of an infinite set is infinite.
The same applies to the thicknesses for infinite sets:

$A \subseteq B \Rightarrow \left| A\right| \le \left| B\right|$

For infinite sets, however, it is possible for a real subset to have the same power as its base set. For example, the natural numbers are a real subset of the integers, but the two sets are equally powerful (namely, countably infinite).
By Cantor's theorem, the power set of a set always more powerful than the set itself:
A finite set with elements has exactly $2^{n}$ subsets.
The number of -elementary subsets of an -elementary (finite) set is ${\tbinom {n}{k}}$ given by the binomial coefficient