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The inclusive or operation in a Boolean algebra. (In ring theory it is used for the exclusive or operation) ~. 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.
Fundamentals. The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original
Cohen's initial development of the concept was for the purpose of analyzing the definition of and social reaction to these subcultures as a social problem. [1] [8] [25] According to Cohen, a moral panic occurs when a "condition, episode, person or group of persons emerges to become defined as a threat to societal values and interests."
The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC axioms except ...
Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be proved from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers. Every infinite set which can be enumerated by natural numbers is the ...
The continuum hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every set, S, of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into S.
A choice function (also called selector or selection) is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f (A) is an element of A. With this concept, the axiom can be stated: Axiom— For any set X of nonempty sets, there exists a choice function f that is defined on X and maps each set of X to an ...