Set theory - Wikipedia
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory — as a branch of mathematics — is mostly concerned with those that …
Set-theoretic definition of natural numbers - Wikipedia
In set theory, several ways have been proposed to construct the natural numbers. These include the representation via von Neumann ordinals, commonly employed in axiomatic set theory, and a system based on equinumerosity that was proposed by Gottlob Frege and by Bertrand Russell.
numbers with a description is a set. In fact it would be better to stay away from using languages like English to describe sets. Our first task will be to build a new language for describing sets, one in which such contradictions cannot arise. We also need to clarify exactly what is meant by “set”. What is a set?
Set Theory - Stanford Encyclopedia of Philosophy
2014年10月8日 · Set theory is the mathematical theory of well-determined collections, called sets, of objects that are called members, or elements, of the set. Pure set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets.
Set theory began with Cantor’s proof in 1874 that the natural numbers do not have the same cardinality as the real numbers. Cantor’s original motivation was to give a new proof of Liouville’s theorem that there are non-algebraic real numbers1. However, Cantor soon began researching set theory for its own sake.
We denote the set [0..n−1] by Zn. x and y are congruent modulo n. We may omit (mod n) when it is clear from context. Every integer x is congruent to some y in Zn. When we add or subtract multiples of n from an integer x to reach some y ∈ Zn, we …
Set Theory – Definition, Types, Operations - GeeksforGeeks
2024年11月18日 · Set theory is a branch of mathematical logic that studies sets, which are collections of objects. These objects are called elements of the set. Learn complete set theory with examples.
Set theory began with Cantor’s proof in 1874 that the natural numbers do not have the same cardinality as the real numbers. Cantor’s original motivation was to give a new proof of Liouville’s theorem that there are non-algebraic real numbers1. However, Cantor soon began researching set theory for its own sake.
Set Theory - Stanford Encyclopedia of Philosophy
2002年7月11日 · Set Theory is the mathematical science of the infinite. It studies properties of sets, abstract objects that pervade the whole of modern mathematics. ... For instance, it is desirable to have the “set of all integers that are divisible by number 3,” the “set of all straight lines in the Euclidean plane that are parallel to a given line ...
Number Theory - SpringerLink
This is an introductory course in number theory. The fundamental theorem of arithmetic is proved from the well-ordering principle, focusing on relative primeness and greatest common divisors. ... Let d be the least number in the non-void set T consisting of all natural numbers of the form \(au+bv\), where u and v vary over the integers. Write \ ...
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