In

File:Example of A is a proper subset of B.svg, A is a proper subset of B
File:Example of C is no proper subset of B.svg, C is a subset but not a proper subset of B

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

''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''. A ''k''-subset is a subset with ''k'' elements.
The subset relation defines a partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...

on sets. In fact, the subsets of a given set form a Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...

under the subset relation, in which the join and meet
In mathematics, specifically order theory, the join of a subset S of a partially ordered set P is the supremum (least upper bound) of S, denoted \bigvee S, and similarly, the meet of S is the infimum (greatest lower bound), denoted \bigwedge S. I ...

are given by intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...

and union, and the subset relation itself is the Boolean inclusion relation.
Definition

If ''A'' and ''B'' are sets and everyelement
Element or elements may refer to:
Science
* Chemical element, a pure substance of one type of atom
* Heating element, a device that generates heat by electrical resistance
* Orbital elements, parameters required to identify a specific orbit of o ...

of ''A'' is also an element of ''B'', then:
:*''A'' is a subset of ''B'', denoted by $A\; \backslash subseteq\; B$, or equivalently,
:* ''B'' is a superset of ''A'', denoted by $B\; \backslash supseteq\; A.$
If ''A'' is a subset of ''B'', but ''A'' is not equal
Equal(s) may refer to:
Mathematics
* Equality (mathematics).
* Equals sign (=), a mathematical symbol used to indicate equality.
Arts and entertainment
* ''Equals'' (film), a 2015 American science fiction film
* ''Equals'' (game), a board game
...

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 $A\; \backslash subsetneq\; B$, or equivalently,
:* ''B'' is a proper (or strict) superset of ''A'', denoted by $B\; \backslash supsetneq\; A$.
The empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...

, written $\backslash $ or $\backslash varnothing,$ is a subset of any set ''X'' and a proper subset of any set except itself, the inclusion relation $\backslash subseteq$ is a partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...

on the set $\backslash mathcal(S)$ (the power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

of ''S''—the set of all subsets of ''S'') defined by $A\; \backslash leq\; B\; \backslash iff\; A\; \backslash subseteq\; B$. We may also partially order $\backslash mathcal(S)$ by reverse set inclusion by defining $A\; \backslash leq\; B\; \backslash text\; B\; \backslash subseteq\; A.$
When quantified, $A\; \backslash subseteq\; B$ is represented as $\backslash forall\; x\; \backslash left(x\; \backslash in\; A\; \backslash implies\; x\; \backslash in\; B\backslash right).$
We can prove the statement $A\; \backslash subseteq\; B$ by applying a proof technique known as the element argument:Let sets ''A'' and ''B'' be given. To prove that $A\; \backslash subseteq\; B,$ # suppose that ''a'' is a particular but arbitrarily chosen element of A # show that ''a'' is an element of ''B''.The validity of this technique can be seen as a consequence of Universal generalization: the technique shows $c\; \backslash in\; A\; \backslash implies\; c\; \backslash in\; B$ for an arbitrarily chosen element ''c''. Universal generalisation then implies $\backslash forall\; x\; \backslash left(x\; \backslash in\; A\; \backslash implies\; x\; \backslash in\; B\backslash right),$ which is equivalent to $A\; \backslash subseteq\; B,$ as stated above. The set of all subsets of $A$ is called its

powerset
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

, and is denoted by $\backslash mathcal(A)$. The set of all $k$-subsets of $A$ is denoted by $\backslash tbinom$, in analogue with the notation for binomial coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the te ...

, which count the number of $k$-subsets of an $n$-element set. In set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

, the notation $;\; href="/html/ALL/s/.html"\; ;"title="">$ is also common, especially when $k$ is a transfinite cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...

.
Properties

* A set ''A'' is a subset of ''B''if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bico ...

their intersection is equal to A.
:Formally:
:$A\; \backslash subseteq\; B\; \backslash text\; A\; \backslash cap\; B\; =\; A.$
* A set ''A'' is a subset of ''B'' if and only if their union is equal to B.
:Formally:
:$A\; \backslash subseteq\; B\; \backslash text\; A\; \backslash cup\; B\; =\; B.$
* A finite set ''A'' is a subset of ''B'', if and only if the cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

of their intersection is equal to the cardinality of A.
:Formally:
:$A\; \backslash subseteq\; B\; \backslash text\; ,\; A\; \backslash cap\; B,\; =\; ,\; A,\; .$
⊂ and ⊃ symbols

Some authors use the symbols $\backslash subset$ and $\backslash supset$ to indicate and respectively; that is, with the same meaning as and instead of the symbols $\backslash subseteq$ and $\backslash supseteq.$ For example, for these authors, it is true of every set ''A'' that $A\; \backslash subset\; A.$ Other authors prefer to use the symbols $\backslash subset$ and $\backslash supset$ to indicate (also called strict) subset and superset respectively; that is, with the same meaning as and instead of the symbols $\backslash subsetneq$ and $\backslash supsetneq.$ This usage makes $\backslash subseteq$ and $\backslash subset$ analogous to theinequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...

symbols $\backslash leq$ and $<.$ For example, if $x\; \backslash leq\; y,$ 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 $\backslash subset$ is proper subset, if $A\; \backslash subseteq\; B,$ then ''A'' may or may not equal ''B'', but if $A\; \backslash subset\; B,$ then ''A'' definitely does not equal ''B''.
Examples of subsets

* The set A = is a proper subset of B = , thus both expressions $A\; \backslash subseteq\; B$ and $A\; \backslash subsetneq\; B$ are true. * The set D = is a subset (but a proper subset) of E = , thus $D\; \backslash subseteq\; E$ is true, and $D\; \backslash subsetneq\; E$ is not true (false). * Any set is a subset of itself, but not a proper subset. ($X\; \backslash subseteq\; X$ is true, and $X\; \backslash subsetneq\; X$ is false for any set X.) * The set is a proper subset of * The set ofnatural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...

s is a proper subset of the set of rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

s; likewise, the set of points in a line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance betwee ...

is a proper subset of the set of points in a line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...

. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

(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
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all r ...

is a proper subset of the set of real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s. 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
An Euler diagram (, ) is a diagrammatic means of representing sets and their relationships. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are similar to another set diagramming technique, Ve ...

:
Other properties of inclusion

Inclusion is the canonicalpartial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...

, in the sense that every partially ordered set $(X,\; \backslash preceq)$ is isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

to some collection of sets ordered by inclusion. The ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...

s are a simple example: if each ordinal ''n'' is identified with the set $;\; href="/html/ALL/s/.html"\; ;"title="">$power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

$\backslash operatorname(S)$ of a set ''S'', the inclusion partial order is—up to an order isomorphism
In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be con ...

—the Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...

of $k\; =\; ,\; S,$ (the cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

of ''S'') copies of the partial order on $\backslash $ for which $0\; <\; 1.$ This can be illustrated by enumerating $S\; =\; \backslash left\backslash ,$, and associating with each subset $T\; \backslash subseteq\; S$ (i.e., each element of $2^S$) the ''k''-tuple from $\backslash ^k,$ of which the ''i''th coordinate is 1 if and only if $s\_i$ is a member of ''T''.
See also

* Convex subset * Inclusion order *Region
In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and the interaction of humanity and ...

* Subset sum problem The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S of integers and a target-sum T, and the question is to decide whether any subset of the integers sum to precisely T''.'' ...

* Subsumptive containment
* Total subset
References

Bibliography

*External links

* * {{Common logical symbols Basic concepts in set theory