Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

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# PowerSet

```A PowerSet is a construct derived from another existing set S.   Given a set S, the PowerSet of S, written P(S), is the set of all subsets of S.  The definition of a subset is as follows: Given sets S and T, then T is defined to be a subset of S if every element of T is also an element of S.

If S is the set {A, B, C} then {A,C} is a subset of S.  There are sets that are also subset of S as well, the complete list is as follows:

* {}
* {A}
* {B}
* {C}
* {A, B}
* {A, C}
* {B, C}
* {A, B, C}

So the PowerSet of S, written P(S), is the set containing all the subsets above.  Written out this would be the set:

P(S) = {  {}, {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C} }

In case you were wondering, for any given set S, S is always considered to be a subset of itself (by definition) - albeit not a ProperSubset.  Also, the empty set, written {}, is also a subset of any given set S.  This is because the empty set vaccuously satisfies the definition of a subset of S; since the empty set has no elements, every element it the empty set is also an element of S, for any given set S.

If the above paragraph sounds a bit like double-speak - it is only because it is stating the obvious.
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