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In mathematics, set ''A'' is a subset of a set ''B'' if all
elements 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'' 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 binary ...
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 i ...
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. ...
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, their ...
and
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...
, and the subset relation itself is the Boolean inclusion relation.


Definition

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, or equivalently, :* ''B'' is a superset of ''A'', denoted by B \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 In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, whe ...
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, or equivalently, :* ''B'' is a proper (or strict) superset of ''A'', denoted by B \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 other ...
, written \ or \varnothing, is a subset of any set ''X'' and a proper subset of any set except itself, the inclusion relation \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 binary ...
on the set \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 post ...
of ''S''—the set of all subsets of ''S'') defined by A \leq B \iff A \subseteq B. We may also partially order \mathcal(S) by reverse set inclusion by defining A \leq B \text B \subseteq A. When quantified, A \subseteq B is represented as \forall x \left(x \in A \implies x \in B\right). We can prove the statement A \subseteq B by applying a proof technique known as the element argument:
Let sets ''A'' and ''B'' be given. To prove that A \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 \in A \implies c \in B for an arbitrarily chosen element ''c''. Universal generalisation then implies \forall x \left(x \in A \implies x \in B\right), which is equivalent to A \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 post ...
, and is denoted by \mathcal(A). The set of all k-subsets of A is denoted by \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 concer ...
, the notation k is also common, especially when k is a
transfinite Transfinite may refer to: * Transfinite number, a number larger than all finite numbers, yet not absolutely infinite * Transfinite induction, an extension of mathematical induction to well-ordered sets ** Transfinite recursion Transfinite inducti ...
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. The ...
.


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 bic ...
their intersection is equal to A. :Formally: : A \subseteq B \text 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 \text A \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 \subseteq B \text , A \cap B, = , A, .


⊂ and ⊃ symbols

Some authors use the symbols \subset and \supset to indicate and respectively; that is, with the same meaning as and instead of the symbols \subseteq and \supseteq. For example, for these authors, it is true of every set ''A'' that A \subset A. Other authors prefer to use the symbols \subset and \supset to indicate (also called strict) subset and superset respectively; that is, with the same meaning as and instead of the symbols \subsetneq and \supsetneq. This usage makes \subseteq and \subset analogous to the inequality symbols \leq and <. For example, if x \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 \subset is proper subset, if A \subseteq B, then ''A'' may or may not equal ''B'', but if A \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 \subseteq B and A \subsetneq B are true. * The set D = is a subset (but a proper subset) of E = , thus D \subseteq E is true, and D \subsetneq E is not true (false). * Any set is a subset of itself, but not a proper subset. (X \subseteq X is true, and X \subsetneq X is false for any set X.) * The set is a proper subset of * The set of
natural 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 ratio ...
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 between ...
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: Art ...
. 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 rat ...
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. Ever ...
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: 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


Other properties of inclusion

Inclusion is the canonical
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 binary ...
, in the sense that every partially ordered set (X, \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 least ...
s are a simple example: if each ordinal ''n'' is identified with the set /math> of all ordinals less than or equal to ''n'', then a \leq b if and only if \subseteq For 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 post ...
\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\tim ...
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 \ for which 0 < 1. This can be illustrated by enumerating S = \left\,, and associating with each subset T \subseteq S (i.e., each element of 2^S) the ''k''-tuple from \^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 In the mathematical field of order theory, an inclusion order is the partial order that arises as the subset-inclusion relation on some collection of objects. In a simple way, every poset ''P'' = (''X'',≤) is (isomorphic to) an inclusion order (j ...
*
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 t ...
* Subset sum problem * Subsumptive containment * Total subset


References


Bibliography

*


External links

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