- You are here
- Everything Explained.Today
- A-Z Contents
- S
- SU
- SUB
- SUBS
- SUBSE
- Subset

In mathematics, a set *A* is a **subset** of a set *B* if all elements of *A* are also elements of *B*; *B* is then a **superset** of *A*. It is possible for *A* and *B* to be equal; if they are unequal, then *A* is a **proper subset** of *B*. The relationship of one set being a subset of another is called **inclusion** (or sometimes **containment**). *A* is a subset of *B* may also be expressed as *B* includes (or contains) *A* or *A* is included (or contained) in *B*.

The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

If *A* and *B* are sets and every element of *A* is also an element of *B*, then:

*A*is a**subset**of*B*, denoted by

*A**\subseteq**B,*

*B*is a**superset**of*A*, denoted by

*B**\supseteq**A.*

If *A* is a subset of *B*, but *A* is not equal to *B* (i.e. there exists at least one element of B which is not an element of *A*), then:

*A*is a**proper**(or**strict**)**subset**of*B*, denoted by

*A**\subsetneq**B*

*B*is a**proper**(or**strict**)**superset**of*A*, denoted by

*B**\supsetneq**A*

- The empty set, written

*\{**\}*

*\varnothing,*

*\subseteq*

l{P}(S)

*A**\leq**B**\iff**A**\subseteq**B*

l{P}(S)

*A**\leq**B*ifandonlyif*B**\subseteq**A.*

When quantified,

*A**\subseteq**B*

*\forall**x**\left(x**\in**A**\implies**x**\in**B\right).*

We can prove the statement

*A**\subseteq**B*

Let setsThe validity of this technique can be seen as a consequence of Universal generalization: the technique showsAandBbe given. To prove that

A\subseteqB,

supposethatais a particular but arbitrarily chosen element ofA,showthatais an element ofB.

*c**\in**A**\implies**c**\in**B*

*\forall**x**\left(x**\in**A**\implies**x**\in**B\right),*

*A**\subseteq**B,*

- A set
*A*is a**subset**of*B*if and only if their intersection is equal to A.

Formally:

*A**\subseteq**B*ifandonlyif*A**\cap**B*=*A.*

- A set
*A*is a**subset**of*B*if and only if their union is equal to B.

Formally:

*A**\subseteq**B*ifandonlyif*A**\cup**B*=*B.*

- A
**finite**set*A*is a**subset**of*B*, if and only if the cardinality of their intersection is equal to the cardinality of A.

Formally:

*A**\subseteq**B*ifandonlyif*|A**\cap**B|*=*|A|.*

Some authors use the symbols

*\subset*

*\supset*

*\subseteq*

*\supseteq.*

*A**\subset**A.*

Other authors prefer to use the symbols

*\subset*

*\supset*

*\subsetneq*

*\supsetneq.*

*\subseteq*

*\subset*

*\leq*

*<.*

*x**\leq**y,*

*x**<**y,*

*\subset*

*A**\subseteq**B,*

*A**\subset**B,*

- The set A = is a proper subset of B =, thus both expressions

*A**\subseteq**B*

*A**\subsetneq**B*

- The set D = is a subset (but a proper subset) of E =, thus

*D**\subseteq**E*

*D**\subsetneq**E*

- Any set is a subset of itself, but not a proper subset. (

*X**\subseteq**X*

*X**\subsetneq**X*

- The set is a proper subset of
- The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition.
- The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or) than the former set.

Another example in an Euler diagram:

Inclusion is the canonical partial order, in the sense that every partially ordered set

*(X,**\preceq)*

*[n]*

*a**\leq**b*

*[a]**\subseteq**[b].*

*\wp{P}(S)*

*k*=*|S|*

*\{*0*,*1*\}*

0*<*1*.*

*S*=*\left\{**s*_{1,}*s*_{2,}*\ldots,**s*_{k}*\right\},*

*T**\subseteq**S*

2^{S}

*\{*0*,*1*\}*^{k,}

*s*_{i}

- Inclusion order
- Region
- Subset sum problem
- Subsumptive containment
- Total subset

- Book: Thomas Jech. Jech, Thomas. Set Theory. Springer-Verlag. 2002. 3-540-44085-2.

- Web site: 2020-04-11. Comprehensive List of Set Theory Symbols. 2020-08-23. Math Vault. en-US.
- Web site: Weisstein. Eric W.. Subset. 2020-08-23. mathworld.wolfram.com. en.
- Book: Rosen, Kenneth H.. Discrete Mathematics and Its Applications. limited. 2012. McGraw-Hill. New York. 978-0-07-338309-5. 119. 7th.
- Book: Epp, Susanna S.. Discrete Mathematics with Applications. 2011. 978-0-495-39132-6. Fourth. 337.