Elementary functionIn mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, log, or x1/n). All elementary functions are continuous on their domains. Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.
TupleIn mathematics, a tuple is a finite sequence or ordered list of numbers or, more generally, mathematical objects, which are called the elements of the tuple. An n-tuple is a tuple of n elements, where n is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called respectively a singleton and an ordered pair. Tuple may be formally defined from ordered pairs by recurrence by starting from ordered pairs; indeed, a n-tuple can be identified with the ordered pair of its (n − 1) first elements and its nth element.
Commutative diagramIn mathematics, and especially in , a commutative diagram is a such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra. A commutative diagram often consists of three parts: (also known as vertices) morphisms (also known as arrows or edges) paths or composites In algebra texts, the type of morphism can be denoted with different arrow usages: A monomorphism may be labeled with a or a .
Inverse functionIn mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by For a function , its inverse admits an explicit description: it sends each element to the unique element such that f(x) = y. As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result.
Identity functionIn mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality f(X) = X is true for all values of X to which f can be applied. Formally, if M is a set, the identity function f on M is defined to be a function with M as its domain and codomain, satisfying In other words, the function value f(X) in the codomain M is always the same as the input element X in the domain M.
Linear functionIn mathematics, the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used. In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map.
Nicolas BourbakiNicolas Bourbaki (nikɔla buʁbaki) is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the Éléments de mathématique (Elements of Mathematics), the group's central work.
Real numberIn mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.
Natural logarithmThe natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
Partial functionIn mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then f is said to be a total function. More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a functional binary relation.
