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In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. [1] In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set . [ 2 ]
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F {\displaystyle F} of subsets of a given set S {\displaystyle S} is called a family of subsets of S {\displaystyle S} , or a family of sets over S ...
Cantor's theorem had immediate and important consequences for the philosophy of mathematics. For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem, we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it.
The Ars Magna (The Great Art, 1545) is an important Latin-language book on algebra written by Gerolamo Cardano. It was first published in 1545 under the title Artis Magnae, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra). There was a second edition in Cardano's lifetime, published in 1570.
This σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets . On the Euclidean space R n , {\displaystyle \mathbb {R} ^{n},} another σ-algebra is of importance: that of all Lebesgue measurable sets.
The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. More specifically, the Millennium Prize version of the conjecture is that, if the elliptic curve E has rank r, then the L-function L(E, s) associated with it vanishes to order r at s = 1.
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
The power set axiom does not specify what subsets of a set exist, only that there is a set containing all those that do. [2] Not all conceivable subsets are guaranteed to exist. In particular, the power set of an infinite set would contain only "constructible sets" if the universe is the constructible universe but in other models of ZF set ...