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Encyclopedia > Subset
Euler diagram showing A is a subset of B
Euler diagram showing
A is a subset of B

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion. Strength training is the use of resistance to muscular contraction to build the strength, endurance and size of skeletal muscles. ... Image File history File links Venn_A_subset_B.svg‎ Venn diagram for A is a subset of B. Modification of Image:Venn A intersect B.svg based on w:en:Image:Venn A subset B.png File links The following pages on the English Wikipedia link to this file (pages on other... Image File history File links Venn_A_subset_B.svg‎ Venn diagram for A is a subset of B. Modification of Image:Venn A intersect B.svg based on w:en:Image:Venn A subset B.png File links The following pages on the English Wikipedia link to this file (pages on other... An Euler diagram does not need to show all possible intersections. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

Contents

Definitions

If A and B are sets and every element of A is also an element of B, then: In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ...

  • A is a subset of (or is included in) B, denoted by A subseteq B,
or equivalently
  • B is a superset of (or includes) 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 not contained in A), then See also the disambiguation page title equality. ...

  • A is also a proper (or strict) subset of B; this is written as Asubsetneq B.
or equivalently
  • B is a proper superset of A; this is written as Bsupsetneq A.

For any set S, the inclusion relation ⊆ is a partial order on the set 2S of all subsets of S (the power set of S). In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...


The symbols ⊂ and ⊃

Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, in place of subsetneq and supsetneq. This usage makes ⊆ and ⊂ analogous to ≤ and < For example, if x ≤ y then x may be equal to y, or maybe not, but if x < y, then x definitely does not equal y, but is strictly less than y. Similarly, using the "⊂ means proper subset" convention, if A ⊆ B, then A may or may not be equal to B, but if A ⊂ B, then A is definitely not equal to B.


Examples

  • The set {1, 2} is a proper subset of {1, 2, 3}.
  • Any set is a subset of itself, but not a proper subset.
  • The empty set, written , is also a subset of any given set X. (This statement is vacuously true, see proof below) The empty set is always a proper subset, except of itself.
  • The set {x : x is a prime number greater than 2000} is a proper subset of {x : x is an odd number greater than 1000}
  • The set of natural numbers is a proper subset of the set of rational numbers and the set of points in a line segment is a proper subset of the set of points in a line. These are counter-intuitive examples in which both the part and the whole are infinite, and the part has the same number of elements as the whole (see Cardinality of infinite sets).

The empty set is the set containing no elements. ... Vacuous truth is a special topic of first-order logic. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... Line redirects here. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ...

Properties

Proposition 1

The empty set is a subset of every set. The empty set is the set containing no elements. ...


Proof: Given any set A, we wish to prove that ø is a subset of A. This involves showing that all elements of ø are elements of A. But there are no elements of ø.


For the experienced mathematician, the inference " ø has no elements, so all elements of ø are elements of A" is immediate, but it may be more troublesome for the beginner. Since ø has no members at all, how can "they" be members of anything else? It may help to think of it the other way around. In order to prove that ø was not a subset of A, we would have to find an element of ø which was not also an element of A. Since there are no elements of ø, this is impossible and hence ø is indeed a subset of A. Vacuous truth is a special topic of first-order logic. ...


Proposition 2

The following proposition says that inclusion is a partial order. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...


If A, B and C are sets then the following hold:

reflexivity: A ⊆ A
antisymmetry: A ⊆ B and B ⊆ A if and only if A = B
transitivity: If A ⊆ B and B ⊆ C then A ⊆ C

In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ... In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ... ↔ ⇔ ≡ logical symbols representing iff. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ...

Proposition 3

The following proposition says that for any set S the power set of S ordered by inclusion is a bounded lattice, and hence together with the distributive and complement laws for unions and intersections (see The fundamental laws of set algebra), show that it is a Boolean algebra. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality (mathematics) and set inclusion. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ...


If A, B and C are subsets of a set S then the following hold:

existence of a least element and a greatest element:
  • ø ⊆ A ⊆ S (that ø ⊆ A is Proposition 1 above.)
existence of joins:
  • A ⊆ AB
  • If A ⊆ C and B ⊆ C then AB ⊆ C
existence of meets:
  • AB ⊆ A
  • If C ⊆ A and C ⊆ B then C ⊆ AB

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ...

Proposition 4

The following proposition says that, the statement "A ⊆ B ", is equivalent to various other statements involving unions, intersections and complements. In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...


For any two sets A and B, the following are equivalent:

  • A ⊆ B
  • A ∩ B  =  A
  • A ∪ B  =  B
  • A − B  =   ø
  • B′ ⊆ A

This shows that the relation of set inclusion can be characterized by either of the set operations of union or intersection, which means that the notion of set inclusion is axiomatically superfluous given either of those operations and equality.


Proposition 5

If the number of elements of the set A is n, then the number of all subsets of A is equal to 2n.


The proof of this is an exercise in induction.


Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (X, preceq) is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set [n] of all ordinals less than or equal to n, then ab if and only if [a] ⊆ [b]. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...


For the power set 2S of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product of k = |S| (the cardinality of S) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating S = {s1, s2, …, sk} and associating with each subset TS (which is to say with each element of 2S) the k-tuple from {0,1}k of which the ith coordinate is 1 if and only if si is a member of T. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... 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. ... In mathematics, the Cartesian product is a direct product of sets. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ...


  Results from FactBites:
 
PubMed Subset Strategies (434 words)
It was created jointly by NLM and the National Cancer Institute to facilitate searching for subjects in all areas of cancer, ranging from clinical care to basic research.
This strategy was developed by NLM and the NASA SPACELINE Office and is intended to retrieve citations of interest to those working in the field of space life sciences research.
This subset is available on the Limits screen under "Subsets." It is also available on the Clinical Queries screen.
Subset - Wikipedia, the free encyclopedia (804 words)
A is a subset of B, and B is a superset of A.
If A is a subset of B, but A is not equal to B, then A is also a proper (or strict) subset of B.
The set of natural numbers is a proper subset of the set of rational numbers.
  More results at FactBites »

 
 

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