# Portal:Set theory

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## Set theory

**Set theory** is the branch of mathematics that studies sets, which are collections of distinct objects. Although any type of objects can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the most well known.

Set theory, formalized using first-order logic, is the most common foundational system for mathematics. The language of set theory is used in the definitions of nearly all mathematical objects, such as functions, and concepts of set theory are integrated throughout the mathematics curriculum. Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects

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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 functions, defined for example in an axiomatic setting. Typically, 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.

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## Selected set

In mathematics, the **Mandelbrot set** is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex *c*-values for which the orbit of 0 under iteration of the complex quadratic polynomial *x*_{n+1}=*x*_{n}^{2} + *c* remains bounded. When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies the boundary as a fractal. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition.

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**Georg Cantor** (March 3, 1845 – January 6, 1918) was a German mathematician. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

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In mathematics, the **symmetric difference** of two sets is the set of elements which are in one of the sets, but not in both. This operation is the set-theoretic kin of the exclusive disjunction (XOR operation) in Boolean logic.

## Did you know?

- ... that there exists a
**composition of binary relations**consistent with the composition of functions ? - ... that there is an
**ordinal arithmetic**extending the arithmetic of natural numbers to the ordinal numbers ? - ... that sets which are both infinite and countable have a cardinality of
**aleph null**?

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*Topics in Set Theory*

Key concepts | Types of set theory | Mathematical logic | Set-theoretic constructions |
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