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Power Set




In Mathematics , given a Set ''S'', the power set (or '''powerset''') of ''S'', written \mathcal{P}(S), ''P''(''S''), or 2''S'' , is the set of all Subset s of ''S''. In Axiomatic Set Theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the Axiom Of Power Set .

Any Subset ''F'' of \mathcal{P}(S) is called a '' Family Of Sets '' over ''S''.
in respect to Inclusion. ]]


AN EXAMPLE

If ''S'' is the set {''x'', ''y'', ''z''}, then the complete list of subsets of ''S'' is as follows:

  • { } (also denoted ∅, the Empty Set )

  • {''x''}

  • {''y''}

  • {''z''}

  • {''x'', ''y''}

  • {''x'', ''z''}

  • {''y'', ''z''}

  • {''x'', ''y'', ''z''}


and hence the power set of ''S'' is

:\mathcal{P}(S) = \left\{\{\}, \{x\}, \{y\}, \{z\}, \{x, y\}, \{x, z\}, \{y, z\}, \{x, y, z\} ight\}\,\!.


PROPERTIES