= Power Set =

A '''power set''' is the [[Analysis/Sets|set]] of all subsets.

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== Description ==

Consider a [[Analysis/Sets|set]] ''A = {x,y,z}''. There exist 8 unique subsets of ''A'', including the empty set and ''A'' itself.
 1. ''{ }'' a.k.a. Ø
 2. ''{x}''
 3. ''{y}''
 4. ''{z}''
 5. ''{x y}''
 6. ''{x z}''
 7. ''{y z}''
 8. ''{x y z}''

The set of these subsets is called the '''power set''' of ''A''. This is notated either as ''𝒫(A)'' (note the calligraphic P), as ''2^A^'', or as ''{B | B ⊆ A}''.

If ''|A| = n'', then ''|𝒫(A)| = 2^n^''. (The second notation above emphasizes this fact.) The [[Analysis/BinomialTheorem|binomial theorem]] provides a clever proof of this. Plug ''a = b = 1'' into the general formula:

{{attachment:binom1.svg}}

{{attachment:binom2.svg}}

'''Cantor's theorem''' proves that ''|A| < |𝒫(A)|'' for any set ''A''.



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CategoryRicottone