# Subset

# Subset

In mathematics, a set *A* is a **subset** of a set *B*, or equivalently *B* is a **superset** of *A*, if *A* is contained in *B*. That is, all elements of *A* are also elements of *B* (note that *A* and *B* may be equal). 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 *A*, or *A* is included 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 meet and join are given by intersection and union.

Definitions

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

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 also a**proper**(or**strict**)**subset**of*B*; this is written as*B*is a**proper superset**of*A*; this is written as

`For any set`

*S*, the inclusionrelation⊆ is apartial orderon the setof all subsets of*S*(thepower setof*S*) defined by. We may also partially orderby reverse set inclusion by definingWhen quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.^{[1]}

Properties

A set

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

- Formally:

A set

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

- Formally:

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:

⊂ and ⊃ symbols

Some authors use the symbols ⊂ and ⊃ to indicate *subset* and *superset* respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇.^{[2]} For example, for these authors, it is true of every set *A* that *A* ⊂ *A*.

Other authors prefer to use the symbols ⊂ and ⊃ to indicate *proper* (also called strict) subset and *proper* superset respectively; that is, with the same meaning and instead of the symbols, ⊊ and ⊋.^{[3]} This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if *x* ≤ *y* then *x* may or may not equal *y*, but if *x* < *y*, then *x* definitely does not equal *y*, and *is* less than *y*. Similarly, using the convention that ⊂ is proper subset, if *A* ⊆ *B*, then *A* may or may not equal *B*, but if *A* ⊂ *B*, then *A* definitely does not equal *B*.

Examples

The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions A ⊆ B and A ⊊ B are true.

The set D = {1, 2, 3} is a subset (but

*not*a proper subset) of E = {1, 2, 3}, thus D ⊆ E is true, and D ⊊ E is not true (false).Any set is a subset of itself, but not a proper subset. (X ⊆ X is true, and X ⊊ X is false for any set X.)

The empty set { }, denoted by ∅, is also a subset of any given set

*X*. It is also always a proper subset of any set except itself.The set {

*x*:*x*is a prime number greater than 10} is a proper subset of {*x*:*x*is an odd number greater than 10}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

*power*) than the former set.

Another example in an Euler diagram:

A is a proper subset of B

C is a subset but not a proper subset of B

Other properties of inclusion

`Inclusion is the canonicalpartial orderin the sense that every partially ordered set (`

*X*,) isisomorphicto some collection of sets ordered by inclusion. Theordinal numbersare a simple example—if each ordinal*n*is identified with the set [*n*] of all ordinals less than or equal to*n*, then*a*≤*b*if and only if [*a*] ⊆ [*b*].`For thepower setof a set`

*S*, the inclusion partial order is (up to anorder isomorphism) theCartesian productof*k*= |*S*| (thecardinalityof*S*) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating*S*= {*s*1,*s*2, ...,*s**k*} and associating with each subset*T*⊆*S*(which is to say with each element of 2*S*) the*k*-tuple from {0,1}*k*of which the*i*th coordinate is 1 if and only if*s**i*is a member of*T*.See also

Containment order

## References

*Discrete Mathematics and Its Applications*(7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5.

*Real and complex analysis*(3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 0924157

*Subsets and Proper Subsets*(PDF), archived from the original (PDF) on 2013-01-23, retrieved 2012-09-07