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Selected article

A set is a collection of distinct objects considered as a whole. Sets are one of the most fundamental concepts in mathematics and their formalization at the end of the 19th century was a major event in the history of mathematics and lead to the unification of a number of different areas. The idea of function comes along naturally, as "morphisms" between sets.

The study of the structure of sets, set theory, can be viewed as a foundational ground for most of mathematical theories. Sets are usually defined axiomatically using an axiomatic set theory, this way to study sets was introduced by Georg Cantor between 1874 and 1884 and deeply inspired later works in logic. Sets are representable in the form of Venn diagrams, for instance it can represent the idea of union, intersection and other operations on sets.

Selected article

In mathematics, a function is a way to assign to each element of a given set exactly one element of another given set. Functions can be abstractly defined in set theory as a functional binary relation between two sets, respectively the domain and the target of the function. There are many ways to give a function, generally using predefined funtions, defined for example in an axiomatic setting. Tipically, functions are expressed by a formula, by a plot or graph, by an algorithm that computes it or by a description of its properties. Sometimes, a function is described through its relationship to other functions (see, for example, inverse function).

One idea of enormous importance in all of mathematics is composition of functions, intuitively: if z is a function of y and y is a function of x, then z is a function of x. The existence of identity functions and some basic properties of functions shows that the class of sets forms a category with functions as morphisms. For a special treatment of functions in a set-theoritical setting, see function (set theory).

Selected article

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite sets. Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers. His proofs, however, give no indication of the extent to which the cardinality of the natural numbers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question. It states:

There is no set whose size is strictly between that of the integers and that of the real numbers.

In light of Cantor's theorem that the sizes of these sets cannot be equal, this hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. The name of the hypothesis comes from the term the continuum for the real numbers.