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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.
According to Bourbaki, the scale of sets on a given set X consists of all sets arising from X by taking Cartesian products and power sets, in any combination, a finite number of times. Examples: X; X × X; P(X); P(P(X × X) × X × P(P(X))) × X. (Here A × B is the product of A and B, and P(A) is the powerset of A.)
Let be the set of ordered pairs of integers (,) with non-zero , and define an equivalence relation on such that (,) (,) if and only if =, then the equivalence class of the pair (,) can be identified with the rational number /, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of ...
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 ...
Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6. If the sets A and B are equal, this is denoted symbolically as A = B (as usual).
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.
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.
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.