Search results
Results from the WOW.Com Content Network
Define the two measures on the real line as = [,] () = [,] for all Borel sets. Then and are equivalent, since all sets outside of [,] have and measure zero, and a set inside [,] is a -null set or a -null set exactly when it is a null set with respect to Lebesgue measure.
Assuming the existence of an infinite set N consisting of all natural numbers and assuming the existence of the power set of any given set allows the definition of a sequence N, P(N), P(P(N)), P(P(P(N))), … of infinite sets where each set is the power set of the set preceding it. By Cantor's theorem, the cardinality of each set in this ...
Ernst Zermelo, a contributer to modern Set theory, was the first to explicitly formalize set equality in his Zermelo set theory (now obsolete), by his Axiom der Bestimmtheit. [31] Equality of sets is axiomatized in set theory in two different ways, depending on whether the axioms are based on a first-order language with or without equality.
Given any set , an equivalence relation over the set [] of all functions can be obtained as follows. Two functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in a permutation.
1. The difference of two sets: x~y is the set of elements of x not in y. 2. An equivalence relation \ The difference of two sets: x\y is the set of elements of x not in y. − The difference of two sets: x−y is the set of elements of x not in y. ≈ Has the same cardinality as × A product of sets / A quotient of a set by an equivalence ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
In set theory, the kernel of a function (or equivalence kernel [1]) may be taken to be either the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function can tell", [2] or; the corresponding partition of the domain.
In set theory, the intersection of two sets and , denoted by , [1] is the set containing all elements of that also belong to or equivalently, all elements of that also belong to . [2] Notation and terminology