Editing Set (mathematics) (section)

==Subsets==
{{Main|Subset}}
A ''subset'' of a set {{tmath|B}} is a set {{tmath|A}} such that every element of {{tmath|A}} is also an element of {{tmath|B}}.<ref name="Hausdorff2005">{{cite book|author=Felix Hausdorff|title=Set Theory|url=https://books.google.com/books?id=yvVIdH16k0YC&pg=PA30|year=2005|publisher=American Mathematical Soc.|isbn=978-0-8218-3835-8|page=30}}</ref> If {{tmath|A}} is a subset of {{tmath|B}}, one says commonly that {{tmath|A}} is ''contained'' in {{tmath|B}},  {{tmath|B}} ''contains'' {{tmath|A}}, or {{tmath|B}} is a ''superset'' of {{tmath|A}}. This denoted {{tmath|A\subseteq B}} and {{tmath|B\supseteq A}}. However many authors use {{tmath|A\subset B}} and {{tmath|B\supset A}} instead. The definition of a subset can be expressed in notation as
<math display=block>A \subseteq B \quad \text{if and only if}\quad \forall x\; (x\in  A \implies x\in B).</math>

A set {{tmath|A}} is a ''proper subset'' of a set {{tmath|B}} if {{tmath|A \subseteq B}} and {{tmath|A\neq B}}. This is denoted {{tmath|A\subset B}} and {{tmath|B\supset A}}. When {{tmath|A\subset B}} is used for the subset relation, or in case of possible ambiguity, one uses commonly {{tmath|A\subsetneq B}} and {{tmath|B\supsetneq A}}.{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/2/mode/2up 3]}}

The [[relation (mathematics)|relationship]] between sets established by ⊆ is called ''inclusion'' or ''containment''. Equality between sets can be expressed in terms of subsets. Two sets are equal if and only if they contain each other: that is, {{math|''A'' ⊆ ''B''}} and {{math|''B'' ⊆ ''A''}} is equivalent to ''A'' = ''B''.<ref name="Lucas1990"/><ref name=":0" /> The empty set is a subset of every set: {{math|∅ ⊆ ''A''}}.{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/8/mode/2up 8]}}

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {{math|{{mset|1, 3}} ⊂ {{mset|1, 2, 3, 4}}}}.
* {{math|{{mset|1, 2, 3, 4}} ⊆ {{mset|1, 2, 3, 4}}}}