Editing Subset (section)

{{short description|Set whose elements all belong to another set}}
{{redirect|Superset}}

{{redirect|⊃|the logic symbol|horseshoe (symbol)|other uses|horseshoe (disambiguation)}}
[[File:Venn A subset B.svg|150px|thumb|right|[[Euler diagram]] showing<br/> ''A'' is a [[subset]] of ''B'' (denoted <math>A \subseteq B</math>) and, conversely, ''B'' is a superset of ''A'' (denoted <math>B \supseteq A</math>).]] 

In mathematics, a [[Set (mathematics)|set]] ''A'' is a '''subset''' of a set ''B'' if all [[Element (mathematics)|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.

When quantified, <math>A \subseteq B</math> is represented as <math>\forall x \left(x \in A \Rightarrow x \in B\right).</math><ref>{{cite book|last=Rosen|first=Kenneth H.|title=Discrete Mathematics and Its Applications|url=https://archive.org/details/discretemathemat00rose_408|url-access=limited|date=2012|publisher=McGraw-Hill|location=New York|isbn=978-0-07-338309-5|page=[https://archive.org/details/discretemathemat00rose_408/page/n139 119]|edition=7th}}</ref>

One can prove the statement <math>A \subseteq B</math> by applying a proof technique known as the element argument<ref>{{Cite book|last=Epp|first=Susanna S.|title=Discrete Mathematics with Applications|year=2011|isbn=978-0-495-39132-6|edition=Fourth|pages=337|publisher=Cengage Learning }}</ref>:<blockquote>Let sets ''A'' and ''B'' be given. To prove that <math>A \subseteq B,</math>

# '''suppose''' that ''a'' is a particular but arbitrarily chosen element of A
# '''show''' that ''a'' is an element of ''B''.
</blockquote>The validity of this technique can be seen as a consequence of [[universal generalization]]: the technique shows <math>(c \in A) \Rightarrow (c \in B)</math> for an arbitrarily chosen element ''c''. Universal generalisation then implies <math>\forall x \left(x \in A \Rightarrow x \in B\right),</math> which is equivalent to <math>A \subseteq B,</math> as stated above.<!-- to allow easy linking to this section which contains math in its name -->