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Subset

From Wikipedia, the free encyclopedia
Set whose elements all belong to another set
"Superset" redirects here. For other uses, seeSuperset (disambiguation).
"⊃" redirects here. For the logic symbol, seeHorseshoe (symbol). For other uses, seeHorseshoe (disambiguation).
Euler diagram showing
A is asubset ofB (denotedA⊆B{\displaystyle A\subseteq B}) and, conversely,B is a superset ofA (denotedB⊇A{\displaystyle B\supseteq A})

In mathematics, asetA is asubset of a setB if allelements ofA are also elements ofB;B is then asuperset ofA. It is possible forA andB to be equal; if they are unequal, thenA is aproper subset ofB. The relationship of one set being a subset of another is calledinclusion (or sometimescontainment).A is a subset ofB may also be expressed asB includes (or contains)A orA is included (or contained) inB. Ak-subset is a subset withk elements.

When quantified,A⊆B{\displaystyle A\subseteq B} is represented as∀x(x∈A⇒x∈B).{\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).}[1]

One can prove the statementA⊆B{\displaystyle A\subseteq B} by applying a proof technique known as the element argument[2]:

Let setsA andB be given. To prove thatA⊆B,{\displaystyle A\subseteq B,}

  1. suppose thata is a particular but arbitrarily chosen element of A
  2. show thata is an element ofB.

The validity of this technique can be seen as a consequence ofuniversal generalization: the technique shows(c∈A)⇒(c∈B){\displaystyle (c\in A)\Rightarrow (c\in B)} for an arbitrarily chosen elementc. Universal generalisation then implies∀x(x∈A⇒x∈B),{\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),} which is equivalent toA⊆B,{\displaystyle A\subseteq B,} as stated above.

Definition

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IfA andB are sets and everyelement ofA is also an element ofB, then:

IfA is a subset ofB, butA is notequal toB (i.e.there exists at least one element of B which is not an element ofA), then:

Theempty set, written{}{\displaystyle \{\}} or∅,{\displaystyle \varnothing ,} has no elements, and therefore isvacuously a subset of any setX.

Basic properties

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A⊆B{\displaystyle A\subseteq B} andB⊆C{\displaystyle B\subseteq C} impliesA⊆C.{\displaystyle A\subseteq C.}

Proper subset

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⊂ and ⊃ symbols

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Some authors use the symbols⊂{\displaystyle \subset } and⊃{\displaystyle \supset } to indicatesubset andsuperset respectively; that is, with the same meaning as and instead of the symbols⊆{\displaystyle \subseteq } and⊇{\displaystyle \supseteq }.[4] For example, for these authors, it is true of every setA thatA⊂A.{\displaystyle A\subset A.} (areflexive relation).

Other authors prefer to use the symbols⊂{\displaystyle \subset } and⊃{\displaystyle \supset } to indicateproper (also called strict) subset andproper superset respectively; that is, with the same meaning as and instead of the symbols⊊{\displaystyle \subsetneq } and⊋.{\displaystyle \supsetneq .}[5] This usage makes⊆{\displaystyle \subseteq } and⊂{\displaystyle \subset } analogous to theinequality symbols≤{\displaystyle \leq } and<.{\displaystyle <.} For example, ifx≤y,{\displaystyle x\leq y,} thenx may or may not equaly, but ifx<y,{\displaystyle x<y,} thenx definitely does not equaly, andis less thany (anirreflexive relation). Similarly, using the convention that⊂{\displaystyle \subset } is proper subset, ifA⊆B,{\displaystyle A\subseteq B,} thenA may or may not equalB, but ifA⊂B,{\displaystyle A\subset B,} thenA definitely does not equalB.

Examples of subsets

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Theregular polygons form a subset of the polygons.
  • The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressionsA⊆B{\displaystyle A\subseteq B} andA⊊B{\displaystyle A\subsetneq B} are true.
  • The set D = {1, 2, 3} is a subset (butnot a proper subset) of E = {1, 2, 3}, thusD⊆E{\displaystyle D\subseteq E} is true, andD⊊E{\displaystyle D\subsetneq E} is not true (false).
  • The set {x:x is aprime number greater than 10} is a proper subset of {x:x is an odd number greater than 10}
  • The set ofnatural numbers is a proper subset of the set ofrational numbers; likewise, the set of points in aline segment is a proper subset of the set of points in aline. These are two examples in which both the subset and the whole set are infinite, and the subset has the samecardinality (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 ofrational numbers is a proper subset of the set ofreal numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (orpower) than the former set.

Another example in anEuler diagram:

  • A is a proper subset of B.
    A is a proper subset of B.
  • C is a subset but not a proper subset of B.
    C is a subset but not a proper subset of B.

Power set

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The set of all subsets ofS{\displaystyle S} is called itspower set, and is denoted byP(S){\displaystyle {\mathcal {P}}(S)}.[6]

The inclusionrelation⊆{\displaystyle \subseteq } is apartial order on the setP(S){\displaystyle {\mathcal {P}}(S)} defined byA≤B⟺A⊆B{\displaystyle A\leq B\iff A\subseteq B}. We may also partially orderP(S){\displaystyle {\mathcal {P}}(S)} by reverse set inclusion by definingA≤B if and only if B⊆A.{\displaystyle A\leq B{\text{ if and only if }}B\subseteq A.}

For the power setP⁡(S){\displaystyle \operatorname {\mathcal {P}} (S)} of a setS, the inclusion partial order is—up to anorder isomorphism—theCartesian product ofk=|S|{\displaystyle k=|S|} (thecardinality ofS) copies of the partial order on{0,1}{\displaystyle \{0,1\}} for which0<1.{\displaystyle 0<1.} This can be illustrated by enumeratingS={s1,s2,…,sk},{\displaystyle S=\left\{s_{1},s_{2},\ldots ,s_{k}\right\},}, and associating with each subsetT⊆S{\displaystyle T\subseteq S} (i.e., each element of2S{\displaystyle 2^{S}}) thek-tuple from{0,1}k,{\displaystyle \{0,1\}^{k},} of which theith coordinate is 1 if and only ifsi{\displaystyle s_{i}} is amember ofT.

The set of allk{\displaystyle k}-subsets ofA{\displaystyle A} is denoted by(Ak){\displaystyle {\tbinom {A}{k}}}, in analogue with the notation forbinomial coefficients, which count the number ofk{\displaystyle k}-subsets of ann{\displaystyle n}-element set. Inset theory, the notation[A]k{\displaystyle [A]^{k}} is also common, especially whenk{\displaystyle k} is atransfinitecardinal number.

Other properties of inclusion

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  • A setA is asubset ofBif and only if their intersection is equal to A. Formally:
A⊆B if and only if A∩B=A.{\displaystyle A\subseteq B{\text{ if and only if }}A\cap B=A.}
  • A setA is asubset ofB if and only if their union is equal to B. Formally:
A⊆B if and only if A∪B=B.{\displaystyle A\subseteq B{\text{ if and only if }}A\cup B=B.}
  • Afinite setA is asubset ofB, if and only if thecardinality of their intersection is equal to the cardinality of A. Formally:
A⊆B if and only if |A∩B|=|A|.{\displaystyle A\subseteq B{\text{ if and only if }}|A\cap B|=|A|.}

See also

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  • Convex subset – In geometry, set whose intersection with every line is a single line segmentPages displaying short descriptions of redirect targets
  • Inclusion order – Partial order that arises as the subset-inclusion relation on some collection of objects
  • Mereology – Study of parts and the wholes they form
  • Region – Connected open subset of a topological spacePages displaying short descriptions of redirect targets
  • Subset sum problem – Decision problem in computer science
  • Subsumptive containment – System of elements that are subordinated to each other
  • Subspace – Mathematical set with some added structurePages displaying short descriptions of redirect targets
  • Total subset – Subset T of a topological vector space X where the linear span of T is a dense subset of X

References

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  1. ^Rosen, Kenneth H. (2012).Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119.ISBN 978-0-07-338309-5.
  2. ^Epp, Susanna S. (2011).Discrete Mathematics with Applications (Fourth ed.). Cengage Learning. p. 337.ISBN 978-0-495-39132-6.
  3. ^Stoll, Robert R. (1963).Set Theory and Logic. San Francisco, CA: Dover Publications.ISBN 978-0-486-63829-4.{{cite book}}:ISBN / Date incompatibility (help)
  4. ^Rudin, Walter (1987),Real and complex analysis (3rd ed.), New York:McGraw-Hill, p. 6,ISBN 978-0-07-054234-1,MR 0924157
  5. ^Subsets and Proper Subsets(PDF), archived fromthe original(PDF) on 2013-01-23, retrieved2012-09-07
  6. ^Weisstein, Eric W."Subset".mathworld.wolfram.com. Retrieved2020-08-23.

Bibliography

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External links

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