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SET THEORY

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0 (number) 0 (zero; BrE: /ˈzɪərəʊ/ or AmE: /ˈziːroʊ/) is both a number[1] and the numerical digit used to represent that number in numerals.

0 (number)

It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems. In the English language, 0 may be called zero, nought or (US) naught /ˈnɔːt/, nil, or — in contexts where at least one adjacent digit distinguishes it from the letter "O" — oh or o /ˈoʊ/. Informal or slang terms for zero include zilch and zip.[2] Ought or aught /ˈɔːt/ has also been used historically.[3] (See Names for the number 0 in English.)

Etymology The word zero came via French zéro from Venetian zero, which (together with cypher) came via Italian zefiro from Arabic صفر, ṣafira = "it was empty", ṣifr = "zero", "nothing". Modern usage History Egypt Ancient Egyptian numerals were base 10. Mesopotamia India. Cardinality. The cardinality of a set A is usually denoted | A |, with a vertical bar on each side; this is the same notation as absolute value and the meaning depends on context.

Cardinality

Alternatively, the cardinality of a set A may be denoted by n(A), A, card(A), or # A. Comparing sets[edit] Definition 1: | A | = | B |[edit] For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as the set N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. Traffic flow (computer networking) A TCP/IP flow can be uniquely identified by the following parameters within a certain time period: Source and Destination IP addressSource and Destination PortLayer 4 Protocol (TCP/UDP/ICMP) All packets with the same source address/port and destination address/port within a time period are considered as one flow.

Traffic flow (computer networking)

Since UDP is uni-directional, it causes one flow. ICMP is bi-directional, so it causes two flows. Establishing a TCP connection begins with a three-way handshake and creates two flows. Flow network. Set theory (music) Georg Cantor. Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔr/ KAN-tor; German: [ˈɡeɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp ˈkantɔʁ]; March 3 [O.S.

Georg Cantor

February 19] 1845 – January 6, 1918[1]) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Giuseppe Peano. Giuseppe Peano (Italian: [dʒuˈzɛppe peˈaːno]; 27 August 1858 – 20 April 1932) was an Italian mathematician.

Giuseppe Peano

The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction. He spent most of his career teaching mathematics at the University of Turin. Biography[edit] Giuseppe Peano and his wife Carola Crosio in 1887. Set theory. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s.

Set theory

After the discovery of paradoxes in naive 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 best-known. Set Theory Basics. Set-builder notation. Direct, ellipses, and informally specified sets[edit] A set is an unordered list of elements.

Set-builder notation

The elements are also called set 'members'. Elements can be any mathematical entity. Ordinal number. Representation of the ordinal numbers up to ωω.

Ordinal number

Each turn of the spiral represents one power of ω In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated. Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and to classify sets with certain kinds of order structures on them.[1] He derived them by accident while working on a problem concerning trigonometric series. Order theory. For a topical guide to this subject, see Outline of order theory.

Order theory

Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions.

A list of order-theoretic terms can be found in the order theory glossary. Background and motivation[edit] Orders are everywhere in mathematics and related fields like computer science. Well-order. Every non-empty well-ordered set has a least element.

Well-order

Every element s of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements besides the least element which have no predecessor (see Natural numbers below for an example). Partially ordered set. Glossary of order theory. Group (mathematics)