power set

Definitions

The set whose elements comprise all the subsets of S (including the empty set and S itself).

Equivalents

Examples

“The power set⟳ of #92;#123;1,2#92;#125; is #92;left#92;#123;#92;empty,#92;#123;1#92;#125;,#92;#123;2#92;#125;,#92;#123;1,2#92;#125;#92;right#92;#125;.”

“Moreover, for notational convenience, we write⟳ the cardinality of a denumerable set⟳ as #92;aleph#95;0. Cardinality of the power set⟳ of a denumerable set⟳ is written as #92;aleph#95;1. We may thus extend⟳ this notation further by taking cardinality of the power set⟳ of the power set⟳ of a denumerable set⟳ as #92;aleph#95;2, etc. but we do not have⟳ the need⟳ for it right now.”

“Theorem 4.1. A complete⟳ Boolean algebra B has a set⟳ of (complete⟳ and atomic) ca-free generators iff B is isomorphic to the power set⟳ of a power set⟳.”

“Exponentiation is essentially a move⟳ to the power set⟳—the set⟳ of all subsets of a given set⟳. This is one of the reasons why Bertrand Russell's paradox is indeed a paradox: We cannot find⟳ a universal set⟳ because no set⟳ can contain⟳ its own⟳ power set⟳!”

CEFR level