Set theory The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. 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 is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. History[edit] Mathematical topics typically emerge and evolve through interactions among many researchers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Cantor's work initially polarized the mathematicians of his day. The next wave of excitement in set theory came around 1900, when it was discovered that Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes.
Arity In logic, mathematics, and computer science, the arity Examples[edit] The term "arity" is rarely employed in everyday usage. For example, rather than saying "the arity of the addition operation is 2" or "addition is an operation of arity 2" one usually says "addition is a binary operation". In general, the naming of functions or operators with a given arity follows a convention similar to the one used for n-based numeral systems such as binary and hexadecimal. One combines a Latin prefix with the -ary ending; for example: A nullary function takes no arguments.A unary function takes one argument.A binary function takes two arguments.A ternary function takes three arguments.An n-ary function takes n arguments. Nullary[edit] Unary[edit] Binary[edit] Most operators encountered in programming are of the binary form. Ternary[edit] with arbitrary precision. n-ary[edit] Variable arity[edit] In computer science, a function accepting a variable number of arguments is called variadic. Other names[edit]
Cartesian product Cartesian product of the sets and The simplest case of a Cartesian product is the Cartesian square, which returns a set from two sets. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). A Cartesian product of n sets can be represented by an array of n dimensions, where each element is an n-tuple. The Cartesian product is named after René Descartes,[1] whose formulation of analytic geometry gave rise to the concept. Examples[edit] A deck of cards[edit] An illustrative example is the standard 52-card deck. Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}. Suits × Ranks returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}. A two-dimensional coordinate system[edit] . , where For example: , or
Signature (logic) In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. Signatures play the same role in mathematics as type signatures in computer programming. Formally, a (single-sorted) signature can be defined as a triple σ = (Sfunc, Srel, ar), where Sfunc and Srel are disjoint sets not containing any other basic logical symbols, called respectively function symbols (examples: +, ×, 0, 1) andrelation symbols or predicates (examples: ≤, ∈), and a function ar: Sfunc Srel → which assigns a non-negative integer called arity to every function or relation symbol. A signature with no function symbols is called a relational signature, and a signature with no relation symbols is called an algebraic signature. "The standard signature for abelian groups is σ = (+,–,0), where – is a unary operator." S.
Model theory This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang and Keisler (1990):[1] universal algebra + logic = model theory. Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997): although model theorists are also interested in the study of fields. In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. Branches of model theory[edit] This article focuses on finitary first order model theory of infinite structures. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. and or
First-order logic A theory about some topic is usually first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions which map from that domain into it, finitely many predicates defined on that domain, and a recursive set of axioms which are believed to hold for those things. Sometimes "theory" is understood in a more formal sense, which is just a set of sentences in first-order logic. The adjective "first-order" distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.[1] In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets. First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Introduction[edit] . x in .
Well-formed formula Introduction[edit] A key use of formulae is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. Although the term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence being expressed, with the marks being a token instance of formula. Propositional calculus[edit] The formulas of propositional calculus, also called propositional formulas,[2] are expressions such as . The formulae are inductively defined as follows: Each propositional variable is, on its own, a formula.If φ is a formula, then φ is a formula.If φ and ψ are formulas, and • is any binary connective, then ( φ • ψ) is a formula. This definition can also be written as a formal grammar in Backus–Naur form, provided the set of variables is finite: <form> ::= <alpha set> | <form>) q) q
Atomic formula Atomic formula in first-order logic[edit] The well-formed terms and propositions of ordinary first-order logic have the following syntax: Terms: that is, a term is recursively defined to be a constant c (a named object from the domain of discourse), or a variable x (ranging over the objects in the domain of discourse), or an n-ary function f whose arguments are terms tk. Propositions: An atomic formula or atom is simply a predicate applied to a tuple of terms; that is, an atomic formula is a formula of the form P (t1, …, tn) for P a predicate, and the tk terms. All other well-formed formulae are obtained by composing atoms with logical connectives and quantifiers. For example, the formula ∀x. When all of the terms in an atom are ground terms, then the atom is called a ground atom or ground predicate. See also[edit] References[edit]
Domain of discourse The term universe of discourse generally refers to the collection of objects being discussed in a specific discourse. In model-theoretical semantics, a universe of discourse is the set of entities that a model is based on. The concept universe of discourse is generally attributed to Augustus De Morgan (1846) but the name was used for the first time in history by George Boole (1854) on page 42 of his Laws of Thought in a long and incisive passage well worth study. Boole's definition is quoted below. The concept, probably discovered independently by Boole in 1847, played a crucial role in his philosophy of logic especially in his stunning principle of wholistic reference. A database is a model of some aspect of the reality of an organisation. Boole’s 1854 Definition[edit] See also[edit] References[edit] Jump up ^ Corcoran, John.
Simple Gates" There are three, five or seven simple gates that you need to learn about, depending on how you want to count them (you will see why in a moment). With these simple gates you can build combinations that will implement any digital component you can imagine. These gates are going to seem a little dry here, and incredibly simple, but we will see some interesting combinations in the following sections that will make them a lot more inspiring. NOT Gate The simplest possible gate is called an "inverter," or a NOT gate. The NOT gate has one input called A and one output called Q ("Q" is used for the output because if you used "O," you would easily confuse it with zero). AND Gate The AND gate performs a logical "and" operation on two inputs, A and B: The idea behind an AND gate is, "If A AND B are both 1, then Q should be 1." 0 0 0 If A is 0 AND B is 0, Q is 0. 0 1 0 If A is 0 AND B is 1, Q is 0. 1 0 0 If A is 1 AND B is 0, Q is 0. 1 1 1 If A is 1 AND B is 1, Q is 1. OR Gate NOR Gate NAND Gate XOR Gate
List of numbers Notable numbers Natural numbers[edit] , Unicode U+2115 ℕ DOUBLE-STRUCK CAPITAL N). The inclusion of 0 in the set of natural numbers is ambiguous and subject to individual definitions. Mathematical significance[edit] Natural numbers may have properties specific to the individual number or may be part of a set (such as prime numbers) of numbers with a particular property. List of mathematically significant natural numbers Cultural or practical significance[edit] Along with their mathematical properties, many integers have cultural significance[2] or are also notable for their use in computing and measurement. List of integers notable for their cultural meanings List of integers notable for their use in units, measurements and scales List of integers notable in computing Classes of natural numbers[edit] Subsets of the natural numbers, such as the prime numbers, may be grouped into sets, for instance based on the divisibility of their members. Prime numbers[edit] The first 100 prime numbers are: