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==Definition==
A set {{M|A}} is a ''subset of'' a set {{M|B}} if {{M|A\subseteq B}}, that is (by the [[implies-subset relation]]):
* {{M|\forall a\in A[a\in B]}} which comes from {{M|1=\forall x[x\in A\implies x\in B]}}.
This may be written {{M|A\in\mathcal{P}(B)}} (where {{M|\mathcal{P}(B)}} denotes the [[power-set]] of {{M|B}}, by definition, all subsets of {{M|B}}!), or {{M|A\subseteq B}}. 

Some authors use {{M|A\subset B}}, however we reserve this for ''proper subset'' - see below. Some authors who use this for what we use {{M|\subseteq}} use {{M|\subsetneq}} for the proper case.
===Proper subset===
A subset is called ''proper'' if {{M|A\subseteq B}} and {{M|A\ne B}}.
==Immediate claims==
# Note that [[emptyset|{{M|\emptyset}}]]{{M|\subseteq A}} for all sets {{M|A}}.
# Note that {{M|\emptyset\subset A}} for all ''[[non-empty]]'' sets {{M|A}}.
==See also==
* [[Partial order]]
* [[Subset relation]]
* [[Lattice Theory (subject)]]
==References==
<references/>
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{{Definition|Elementary Set Theory}}