Editing Subset (section)

==Power set==
The set of all subsets of <math>S</math> is called its [[power set]], and is denoted by <math>\mathcal{P}(S)</math>.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Subset|url=https://mathworld.wolfram.com/Subset.html|access-date=2020-08-23|website=mathworld.wolfram.com|language=en}}</ref>

The inclusion [[Binary relation|relation]] <math>\subseteq</math> is a [[partial order]] on the set <math>\mathcal{P}(S)</math> defined by <math>A \leq B \iff A \subseteq B</math>. We may also partially order <math>\mathcal{P}(S)</math> by reverse set inclusion by defining <math>A \leq B  \text{ if and only if }  B \subseteq A.</math>

For the power set <math>\operatorname{\mathcal{P}}(S)</math> of a set ''S'', the inclusion partial order is—up to an [[order isomorphism]]—the [[Cartesian product]] of <math>k = |S|</math> (the [[cardinality]] of ''S'') copies of the partial order on <math>\{0, 1\}</math> for which <math>0 < 1.</math> This can be illustrated by enumerating <math>S = \left\{ s_1, s_2, \ldots, s_k \right\},</math>, and associating with each subset <math>T \subseteq S</math> (i.e., each element of <math>2^S</math>) the ''k''-tuple from <math>\{0, 1\}^k,</math> of which the ''i''th coordinate is 1 if and only if <math>s_i</math> is a [[set membership|member]] of ''T''.

The set of all <math>k</math>-subsets of <math>A</math> is denoted by <math>\tbinom{A}{k}</math>, in analogue with the notation for [[binomial coefficients]], which count the number of <math>k</math>-subsets of an <math>n</math>-element set. In [[set theory]], the notation <math>[A]^k</math> is also common, especially when <math>k</math> is a [[transfinite number|transfinite]] [[cardinal number]].