1.
Function (mathematics)
–
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
2.
Domain of a function
–
In mathematics, and more specifically in naive set theory, the domain of definition of a function is the set of input or argument values for which the function is defined. That is, the function provides an output or value for each member of the domain, conversely, the set of values the function takes on as output is termed the image of the function, which is sometimes also referred to as the range of the function. For instance, the domain of cosine is the set of all real numbers, if the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the X-axis. Given a function f, X→Y, the set X is the domain of f, in the expression f, x is the argument and f is the value. One can think of an argument as a member of the domain that is chosen as an input to the function, the image of f is the set of all values assumed by f for all possible x, this is the set. The image of f can be the set as the codomain or it can be a proper subset of it. It is, in general, smaller than the codomain, it is the whole codomain if, a well-defined function must map every element of its domain to an element of its codomain. For example, the function f defined by f =1 / x has no value for f, thus, the set of all real numbers, R, cannot be its domain. In cases like this, the function is defined on R\ or the gap is plugged by explicitly defining f. If we extend the definition of f to f = {1 / x x ≠00 x =0 then f is defined for all real numbers, any function can be restricted to a subset of its domain. The restriction of g, A → B to S, where S ⊆ A, is written g |S, S → B. The natural domain of a function is the set of values for which the function is defined, typically within the reals. For instance the natural domain of square root is the non-negative reals when considered as a real number function, when considering a natural domain, the set of possible values of the function is typically called its range. There are two meanings in current mathematical usage for the notion of the domain of a partial function from X to Y, i. e. a function from a subset X of X to Y. Most mathematicians, including recursion theorists, use the domain of f for the set X of all values x such that f is defined. But some, particularly category theorists, consider the domain to be X, in category theory one deals with morphisms instead of functions. Morphisms are arrows from one object to another, the domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned or at least formulated more abstractly, for example, the notion of restricting a morphism to a subset of its domain must be modified
3.
Codomain
–
In mathematics, the codomain or target set of a function is the set Y into which all of the output of the function is constrained to fall. It is the set Y in the f, X → Y. The codomain is also referred to as the range but that term is ambiguous as it may also refer to the image. The set F is called the graph of the function, the set of all elements of the form f, where x ranges over the elements of the domain X, is called the image of f. In general, the image of a function is a subset of its codomain, thus, it may not coincide with its codomain. Namely, a function that is not surjective has elements y in its codomain for which the equation f = y does not have a solution. An alternative definition of function by Bourbaki, namely as just a functional graph, for example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple. With such a definition functions do not have a codomain, although some still use it informally after introducing a function in the form f, X → Y. For a function f, R → R defined by f, x ↦ x 2, or equivalently f = x 2, the codomain of f is R, but f does not map to any negative number. Thus the image of f is the set R0 +, i. e. the interval [0, an alternative function g is defined thus, g, R → R0 + g, x ↦ x 2. While f and g map a given x to the number, they are not, in this view. A third function h can be defined to demonstrate why, h, x ↦ x, the domain of h must be defined to be R0 +, h, R0 + → R. The compositions are denoted h ∘ f, h ∘ g, on inspection, h ∘ f is not useful. The codomain affects whether a function is a surjection, in that the function is surjective if, in the example, g is a surjection while f is not. The codomain does not affect whether a function is an injection, each matrix represents a map with the domain R2 and codomain R2. Some transformations may have image equal to the codomain but many do not. Take for example the matrix T given by T = which represents a linear transformation that maps the point to, the point is not in the image of T, but is still in the codomain since linear transformations from R2 to R2 are of explicit relevance. Just like all 2×2 matrices, T represents a member of that set, examining the differences between the image and codomain can often be useful for discovering properties of the function in question
4.
Integer-valued function
–
In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain, floor and ceiling functions are examples of an integer-valued function of a real variable, but on real numbers and generally, on topological spaces integer-valued functions are not especially useful. Any such function on a connected space either has discontinuities or is constant, on the other hand, on discrete and other totally disconnected spaces integer-valued functions have roughly the same importance as real-valued functions have on non-discrete spaces. Any function with natural, or non-negative integer values is a case of integer-valued function. Integer-valued functions defined on the domain of all real numbers include the floor and ceiling functions, the Dirichlet function, the sign function, integer-valued functions defined on the domain of non-negative real numbers include the integer square root function and the prime-counting function. On an arbitrary set X, integer-valued functions form a ring with pointwise operations of addition and multiplication, since the latter is an ordered ring, the functions form a partially ordered ring, f ≤ g ⟺ ∀ x, f ≤ g. Integer-valued functions are ubiquitous in graph theory and they also have similar uses in geometric group theory, where length function represents the concept of norm, and word metric represents the concept of metric. Integer-valued polynomials are important in ring theory, in mathematical logic such concepts as a primitive recursive function and a μ-recursive function represent integer-valued functions of several natural variables or, in other words, functions on Nn. Gödel numbering, defined on well-formed formulae of some language, is a natural-valued function. Computability theory is based on natural numbers and natural functions on them. In number theory, many functions are integer-valued. In computer programming many functions return values of integer type due to simplicity of implementation, integer-valued polynomial Semi-continuity Rank #Mathematics Grade #In mathematics
5.
Sequence
–
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members, the number of elements is called the length of the sequence. Unlike a set, order matters, and exactly the elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index and it depends on the context or of a specific convention, if the first element has index 0 or 1. For example, is a sequence of letters with the letter M first, also, the sequence, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, the empty sequence is included in most notions of sequence, but may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order, Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations, Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are useful for specific types of sequences. One way to specify a sequence is to list the elements, for example, the first four odd numbers form the sequence. This notation can be used for sequences as well. For instance, the sequence of positive odd integers can be written. Listing is most useful for sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples, the prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence, the prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is, for a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences
6.
Real-valued function
–
In mathematics, a real-valued function or real function is a function whose values are real numbers. In other words, it is a function that assigns a number to each member of its domain. Many important function spaces are defined to consist of real functions, let X be an arbitrary set. Let F denote the set of all functions from X to real numbers R. F is an ordered ring. The σ-algebra of Borel sets is an important structure on real numbers, if X has its σ-algebra and a function f is such that the preimage f −1 of any Borel set B belongs to that σ-algebra, then f is said to be measurable. Measurable functions also form a space and an algebra as explained above. Moreover, a set of real-valued functions on X can actually define a σ-algebra on X generated by all preimages of all Borel sets and this is the way how σ-algebras arise in probability theory, where real-valued functions on the sample space Ω are real-valued random variables. Real numbers form a space and a complete metric space. Continuous real-valued functions are important in theories of topological spaces and of metric spaces, the extreme value theorem states that for any real continuous function on a compact space its global maximum and minimum exist. The concept of space itself is defined with a real-valued function of two variables, the metric, which is continuous. The space of functions on a compact Hausdorff space has a particular importance. Convergent sequences also can be considered as real-valued continuous functions on a topological space. Continuous functions also form a space and an algebra as explained above. Real numbers are used as the codomain to define smooth functions, a domain of a real smooth function can be the real coordinate space, a topological vector space, an open subset of them, or a smooth manifold. Spaces of smooth functions also are vector spaces and algebras as explained above, a measure on a set is a non-negative real-valued functional on a σ-algebra of subsets. Lp spaces on sets with a measure are defined from aforementioned real-valued measurable functions, though, real-valued Lp spaces still have some of the structure explicated above. For example, pointwise product of two L2 functions belongs to L1, Real analysis Partial differential equations, a major user of real-valued functions Norm Scalar Weisstein, Eric W. Real Function
7.
Function of a real variable
–
The output, also called the value of the function, could be anything, simple examples include a single real number, or a vector of real numbers. Vector-valued functions of a real variable occur widely in applied mathematics and physics, particularly in classical mechanics of particles. But we could also have a matrix of numbers as the output. The output could also be other number fields, such as numbers, quaternions. For simplicity, in this article a real-valued function of a variable will be simply called a function. To avoid any ambiguity, the types of functions that may occur will be explicitly specified. In other words, a function of a real variable is a function f, X → R such that its domain X is a subset of ℝ that contains an open set. A simple example of a function in one variable could be, the image of a function f is the set of all values of f when the variable x runs in the whole domain of f. For a continuous real-valued function with a domain, the image is either an interval or a single value. In the latter case, the function is a constant function, the preimage of a given real number y is the set of the solutions of the equation y = f. The domain of a function of real variables is a subset of ℝ that is sometimes. In fact, if one restricts the domain X of a function f to a subset Y ⊂ X, one gets formally a different function, the restriction of f to Y, which is denoted f|Y. In practice, it is not harmful to identify f and f|Y. Conversely, it is possible to enlarge naturally the domain of a given function. This means that it is not worthy to explicitly define the domain of a function of a real variable, the arithmetic operations may be applied to the functions in the following way, For every real number r, the constant function ↦ r, is everywhere defined. For every real number r and every function f, the function r f, ↦ r f has the domain as f. One may similarly define 1 / f, ↦1 / f and this constraint implies that the above two algebras are not fields. Until the second part of 19th century, only continuous functions were considered by mathematicians, as continuous functions of a real variable are ubiquitous in mathematics, it is worth defining this notion without reference to the general notion of continuous maps between topological space
8.
Function of several real variables
–
This concept extends the idea of a function of a real variable to several variables. The input variables take real values, while the output, also called the value of the function, the domain of a function of several variables is the subset of ℝn for which the function is defined. As usual, the domain of a function of real variables is supposed to contain an open subset of ℝn. A real-valued function of n variables is a function that takes as input n real numbers, commonly represented by the variables x1. Xn, for producing another number, the value of the function. For simplicity, in this article a real-valued function of real variables will be simply called a function. To avoid any ambiguity, the types of functions that may occur will be explicitly specified. In other words, a function of n real variables is a function f, X → R such that its domain X is a subset of ℝn that contains an open set. An element of X being an n-tuple, the notation for denoting functions would be f. The common usage, much older than the definition of functions between sets, it to not use double parentheses and to simply write f. It is also common to abbreviate the n-tuple by using a similar to that for vectors, like boldface x, underline x. The domain restricts all variables to be positive since lengths and areas must be positive, for an example of a function in two variables, z, R2 → R z = a x + b y where a and b are real non-zero constants. The function is well-defined at all points in ℝ2. The previous example can be extended easily to higher dimensions, z, R p → R z = a 1 x 1 + a 2 x 2 + ⋯ + a p x p for p non-zero real constants a1, ap, which describes a p-dimensional hyperplane. The Euclidean norm, f = ∥ x ∥ = x 12 + ⋯ + x n 2 is also a function of n variables which is everywhere defined, the function does not include the origin =, if it did then f would be ill-defined at that point. Using a 3d Cartesian coordinate system with the xy plane as the domain ℝ2, and the z axis the codomain ℝ, the image of a function f is the set of all values of f when the n-tuple runs in the whole domain of f. For a continuous real-valued function which has a domain, the image is either an interval or a single value. In the latter case, the function is a constant function, the preimage of a given real number y is called a level set
9.
Complex-valued function
–
In mathematics, a complex-valued function is a function whose values are complex numbers. In other words, it is a function that assigns a number to each member of its domain. This domain does not necessarily have any structure related to complex numbers, most important uses of such functions in complex analysis and in functional analysis are explicated below. A vector space and an algebra of functions over complex numbers can be defined in the same way as for real-valued functions. Complex analysis considers holomorphic functions on manifolds, such as Riemann surfaces. The property of analytic continuation makes them very dissimilar from smooth functions, namely, if a function defined in a neighborhood can be continued to a wider domain, then this continuation is unique. As real functions, any function is infinitely smooth and analytic. But there is much freedom in construction of a holomorphic function than in one of a smooth function. Complex-valued L2 spaces on sets with a measure have a particular importance because they are Hilbert spaces and they often appear in functional analysis and operator theory. A major user of such spaces is quantum mechanics, as wave functions, the sets on which the complex-valued L2 is constructed have the potential to be more exotic than their real-valued analog. Also, complex-valued continuous functions are an important example in the theory of C*-algebras, Function of a complex variable, the dual concept Weisstein, Eric W. Complex Function
10.
Complex analysis
–
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. As a differentiable function of a variable is equal to the sum of its Taylor series. Complex analysis is one of the branches in mathematics, with roots in the 19th century. Important mathematicians associated with complex analysis include Euler, Gauss, Riemann, Cauchy, Weierstrass, Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has very popular through a new boost from complex dynamics. Another important application of analysis is in string theory which studies conformal invariants in quantum field theory. A complex function is one in which the independent variable and the dependent variable are complex numbers. More precisely, a function is a function whose domain. In other words, the components of the f, u = u and v = v can be interpreted as real-valued functions of the two real variables, x and y. The basic concepts of complex analysis are often introduced by extending the elementary real functions into the complex domain, holomorphic functions are complex functions, defined on an open subset of the complex plane, that are differentiable. In the context of analysis, the derivative of f at z 0 is defined to be f ′ = lim z → z 0 f − f z − z 0, z ∈ C. Although superficially similar in form to the derivative of a real function, in particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach z 0 in the complex plane. Consequently, complex differentiability has much stronger consequences than usual differentiability, for instance, holomorphic functions are infinitely differentiable, whereas most real differentiable functions are not. For this reason, holomorphic functions are referred to as analytic functions. Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions z ↦ ℜ, z ↦ | z |, an important property that characterizes holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy-Riemann conditions. If f, C → C, defined by f = f = u + i v, here, the differential operator ∂ / ∂ z ¯ is defined as. In terms of the real and imaginary parts of the function, u and v, this is equivalent to the pair of equations u x = v y and u y = − v x, where the subscripts indicate partial differentiation
11.
Constant function
–
In mathematics, a constant function is a function whose value is the same for every input value. For example, the function y =4 is a constant function because the value of y is 4 regardless of the value x. As a real-valued function of an argument, a constant function has the general form y = c or just y = c. Example, The function y =2 or just y =2 is the constant function where the output value is c =2. The domain of function is the set of all real numbers ℝ. The codomain of this function is just, the independent variable x does not appear on the right side of the function expression and so its value is vacuously substituted. No matter what value of x is input, the output is 2, real-world example, A store where every item is sold for the price of 1 euro. The graph of the constant function y = c is a line in the plane that passes through the point. In the context of a polynomial in one variable x, the constant function is a polynomial of degree 0. This function has no point with the x-axis, that is. On the other hand, the polynomial f =0 is the zero function. It is the constant function and every x is a root and its graph is the x-axis in the plane. A constant function is a function, i. e. the graph of a constant function is symmetric with respect to the y-axis. In the context where it is defined, the derivative of a function is a measure of the rate of change of values with respect to change in input values. Because a constant function does not change, its derivative is 0 and this is often written, ′ =0. Namely, if y=0 for all numbers x, then y is a constant function. Example, Given the constant function y = −2, the derivative of y is the identically zero function y ′ = ′ =0. Every constant function whose domain and codomain are the same is idempotent, every constant function between topological spaces is continuous
12.
Identity function
–
In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f = x, formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies f = x for all elements x in M. In other words, the value f in M is always the same input element x of M. The identity function on M is clearly a function as well as a surjective function. The identity function f on M is often denoted by idM, in set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of M. If f, M → N is any function, then we have f ∘ idM = f = idN ∘ f, in particular, idM is the identity element of the monoid of all functions from M to M. Since the identity element of a monoid is unique, one can define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, the identity function is a linear operator, when applied to vector spaces. The identity function on the integers is a completely multiplicative function. In an n-dimensional vector space the identity function is represented by the identity matrix In, in a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the group only containing this isometry. In a topological space, the identity function is always continuous
13.
Linear map
–
In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE
14.
Polynomial
–
In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate x is x2 − 4x +7, an example in three variables is x3 + 2xyz2 − yz +1. Polynomials appear in a variety of areas of mathematics and science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra, the word polynomial joins two diverse roots, the Greek poly, meaning many, and the Latin nomen, or name. It was derived from the binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century, the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a symbol which does not have any value. It is thus correct to call it an indeterminate. However, when one considers the function defined by the polynomial, then x represents the argument of the function, many authors use these two words interchangeably. It is a convention to use uppercase letters for the indeterminates. However one may use it over any domain where addition and multiplication are defined, in particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. This equality allows writing let P be a polynomial as a shorthand for let P be a polynomial in the indeterminate x. A polynomial is an expression that can be built from constants, the word indeterminate means that x represents no particular value, although any value may be substituted for it. The mapping that associates the result of substitution to the substituted value is a function. This can be expressed concisely by using summation notation, ∑ k =0 n a k x k That is. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, because x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called, respectively, a constant term, the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the polynomial,0, is generally treated as not defined
15.
Rational function
–
In mathematics, a rational function is any function which can be defined by a rational fraction, i. e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be numbers, they may be taken in any field K. In this case, one speaks of a function and a rational fraction over K. The values of the variables may be taken in any field L containing K, then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L. The set of functions over a field K is a field. A function f is called a function if and only if it can be written in the form f = P Q where P and Q are polynomials in x and Q is not the zero polynomial. The domain of f is the set of all points x for which the denominator Q is not zero and it is a common usage to identify f and f 1, that is to extend by continuity the domain of f to that of f 1. Indeed, one can define a rational fraction as a class of fractions of polynomials. In this case P Q is equivalent to P1 Q1, a proper rational function is a rational function in which the degree of P is no greater than the degree of Q and both are real polynomials. The rational function f = x 3 −2 x 2 is not defined at x 2 =5 ⇔ x = ±5 and it is asymptotic to x 2 as x approaches infinity. A constant function such as f = π is a function since constants are polynomials. Note that the function itself is rational, even though the value of f is irrational for all x, every polynomial function f = P is a rational function with Q =1. A function that cannot be written in form, such as f = sin , is not a rational function. The adjective irrational is not generally used for functions, the rational function f = x x is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient of two functions is itself a rational function. However, the process of reduction to standard form may result in the removal of such singularities unless care is taken. Using the definition of functions as equivalence classes gets around this. For example,1 x 2 − x +2 = ∑ k =0 ∞ a k x k, combining like terms gives 1 =2 a 0 + x + ∑ k =2 ∞ x k
16.
Algebraic function
–
In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Examples of such functions are, f =1 / x f = x f =1 + x 3 x 3 /7 −7 x 1 /3 Some algebraic functions, however, cannot be expressed by such finite expressions. This is the case, for example, of the Bring radical, quite often, S = Q, and one then talks about function algebraic over Q, and the evaluation at a given rational value of such an algebraic function gives an algebraic number. A function which is not algebraic is called a function, as it is for example the case of exp , tan . A composition of functions can give an algebraic function, f = cos =1 − x 2. As an equation of degree n has n roots, a polynomial equation does not implicitly define a single function, consider for example the equation of the unit circle, y 2 + x 2 =1. This determines y, except only up to a sign, accordingly. An algebraic function in m variables is similarly defined as a function y which solves an equation in m +1 variables. It is normally assumed that p should be an irreducible polynomial, the existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, a function in m variables over the field K is an element of the algebraic closure of the field of rational functions K. The informal definition of an algebraic function provides a number of clues about their properties and this is something of an oversimplification, because of the fundamental theorem of Galois theory, algebraic functions need not be expressible by radicals. First, note that any polynomial function y = p is an algebraic function, more generally, any rational function y = p q is algebraic, being the solution to q y − p =0. Moreover, the nth root of any polynomial y = p n is an algebraic function, surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solution to a n y n + ⋯ + a 0 =0, for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of x and y and gathering terms, writing x as a function of y gives the inverse function, also an algebraic function. However, not every function has an inverse, for example, y = x2 fails the horizontal line test, it fails to be one-to-one. The inverse is the function x = ± y. Another way to understand this, is that the set of branches of the equation defining our algebraic function is the graph of an algebraic curve
17.
Smoothness
–
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain, differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives, consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer, the function f is said to be of class Ck if the derivatives f′, f′′. The function f is said to be of class C∞, or smooth, if it has derivatives of all orders. The function f is said to be of class Cω, or analytic, if f is smooth, Cω is thus strictly contained in C∞. Bump functions are examples of functions in C∞ but not in Cω, to put it differently, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous, thus, a C1 function is exactly a function whose derivative exists and is of class C0. In particular, Ck is contained in Ck−1 for every k, C∞, the class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers. The function f = { x if x ≥0,0 if x <0 is continuous, because cos oscillates as x →0, f ’ is not continuous at zero. Therefore, this function is differentiable but not of class C1, the functions f = | x | k +1 where k is even, are continuous and k times differentiable at all x. But at x =0 they are not times differentiable, so they are of class Ck, the exponential function is analytic, so, of class Cω. The trigonometric functions are also analytic wherever they are defined, the function f is an example of a smooth function with compact support. Let n and m be some positive integers, if f is a function from an open subset of Rn with values in Rm, then f has component functions f1. Each of these may or may not have partial derivatives, the classes C∞ and Cω are defined as before. These criteria of differentiability can be applied to the functions of a differential structure. The resulting space is called a Ck manifold, if one wishes to start with a coordinate-independent definition of the class Ck, one may start by considering maps between Banach spaces. A map from one Banach space to another is differentiable at a point if there is a map which approximates it at that point
18.
Continuous function
–
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function, a continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the concepts of topology. The introductory portion of this focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the general case of functions between two metric spaces. In order theory, especially in theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article, as an example, consider the function h, which describes the height of a growing flower at time t. By contrast, if M denotes the amount of money in an account at time t, then the function jumps at each point in time when money is deposited or withdrawn. A form of the definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of quantities. The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasnt published until the 1930s, all three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of continuity in 1872. This is not a definition of continuity since the function f =1 x is continuous on its whole domain of R ∖ A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity, a function is then continuous if it has no holes or jumps, that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value, there are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a function is continuous or not. In the definitions below, f, I → R. is a function defined on a subset I of the set R of real numbers and this subset I is referred to as the domain of f
19.
Measurable function
–
In mathematics, particularly in measure theory, a measurable function is a structure-preserving function between measurable spaces. For example, the notion of integrability can be defined for a real-valued measurable function on a measurable space, a measurable function is said to be bimeasurable if it is bijective and its inverse is also measurable. For example, in probability theory, a function on a probability space is known as a random variable. In contrast, functions that are not Lebesgue measurable are generally considered pathological, let and be measurable spaces, meaning that X and Y are sets equipped with respective σ -algebras Σ and T. A function f, X → Y is said to be if the preimage of E under f is in Σ for every E ∈ T, i. e. f −1, = ∈ Σ, ∀ E ∈ T. The notion of measurability depends on the sigma algebras Σ and T, to emphasize this dependency, if f, X → Y is a measurable function, we will write f, →. This definition can be simple, however, as special care must be taken regarding the σ -algebras involved. Here, L is the σ -algebra of Lebesgue measurable sets, and B is the Borel algebra on R, as a result, the composition of Lebesgue-measurable functions need not be Lebesgue-measurable. By convention a topological space is assumed to be equipped with the Borel algebra unless otherwise specified, most commonly this space will be the real or complex numbers. For instance, a measurable function is a function for which the preimage of each Borel set is measurable. A complex-valued measurable function is defined analogously, in practice, some authors use measurable functions to refer only to real-valued measurable functions with respect to the Borel algebra. If the values of the lie in an infinite-dimensional vector space instead of R or C, usually other definitions of measurability are used, such as weak measurability. If and are Borel spaces, a function f, → is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous, however, a measurable function is nearly a continuous function, see Luzins theorem. If a Borel function happens to be a section of some map Y → π X, it is called a Borel section. A Lebesgue measurable function is a function f, →, where L is the σ -algebra of Lebesgue measurable sets. Lebesgue measurable functions are of interest in analysis because they can be integrated. In the case f, X → R, f is Lebesgue measurable iff = is measurable for all α ∈ R and this is also equivalent to any of, being measurable for all α
20.
Injective function
–
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness, it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence, occasionally, an injective function from X to Y is denoted f, X ↣ Y, using an arrow with a barbed tail. A function f that is not injective is sometimes called many-to-one, however, the injective terminology is also sometimes used to mean single-valued, i. e. each argument is mapped to at most one value. A monomorphism is a generalization of a function in category theory. Let f be a function whose domain is a set X, the function f is said to be injective provided that for all a and b in X, whenever f = f, then a = b, that is, f = f implies a = b. Equivalently, if a ≠ b, then f ≠ f, in particular the identity function X → X is always injective. If the domain X = ∅ or X has only one element, the function f, R → R defined by f = 2x +1 is injective. The function g, R → R defined by g = x2 is not injective, however, if g is redefined so that its domain is the non-negative real numbers [0, +∞), then g is injective. The exponential function exp, R → R defined by exp = ex is injective, the natural logarithm function ln, → R defined by x ↦ ln x is injective. The function g, R → R defined by g = xn − x is not injective, since, for example, g = g =0. More generally, when X and Y are both the real line R, then a function f, R → R is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the line test. Functions with left inverses are always injections and that is, given f, X → Y, if there is a function g, Y → X such that, for every x ∈ X g = x then f is injective. In this case, g is called a retraction of f, conversely, f is called a section of g. Conversely, every injection f with non-empty domain has an inverse g. Note that g may not be an inverse of f because the composition in the other order, f o g. In other words, a function that can be undone or reversed, injections are reversible but not always invertible
21.
Surjective function
–
It is not required that x is unique, the function f may map one or more elements of X to the same element of Y. The French prefix sur means over or above and relates to the fact that the image of the domain of a surjective function completely covers the functions codomain, any function induces a surjection by restricting its codomain to its range. Every surjective function has an inverse, and every function with a right inverse is necessarily a surjection. The composite of surjective functions is always surjective, any function can be decomposed into a surjection and an injection. A surjective function is a function whose image is equal to its codomain, equivalently, a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with f = y. Surjections are sometimes denoted by a two-headed rightwards arrow, as in f, X ↠ Y, symbolically, If f, X → Y, then f is said to be surjective if ∀ y ∈ Y, ∃ x ∈ X, f = y. For any set X, the identity function idX on X is surjective, the function f, Z → defined by f = n mod 2 is surjective. The function f, R → R defined by f = 2x +1 is surjective, because for every real number y we have an x such that f = y, an appropriate x is /2. However, this function is not injective since e. g. the pre-image of y =2 is, the function g, R → R defined by g = x2 is not surjective, because there is no real number x such that x2 = −1. However, the g, R → R0+ defined by g = x2 is surjective because for every y in the nonnegative real codomain Y there is at least one x in the real domain X such that x2 = y. The natural logarithm ln, → R is a surjective. Its inverse, the function, is not surjective as its range is the set of positive real numbers. The matrix exponential is not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as a map from the space of all n×n matrices to the linear group of degree n, i. e. the group of all n×n invertible matrices. Under this definition the matrix exponential is surjective for complex matrices, the projection from a cartesian product A × B to one of its factors is surjective unless the other factor is empty. In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function, a function is bijective if and only if it is both surjective and injective. If a function is identified with its graph, then surjectivity is not a property of the function itself, unlike injectivity, surjectivity cannot be read off of the graph of the function alone. The function g, Y → X is said to be an inverse of the function f, X → Y if f = y for every y in Y
22.
Bijection
–
In mathematical terms, a bijective function f, X → Y is a one-to-one and onto mapping of a set X to a set Y. A bijection from the set X to the set Y has a function from Y to X. If X and Y are finite sets, then the existence of a means they have the same number of elements. For infinite sets the picture is complicated, leading to the concept of cardinal number. A bijective function from a set to itself is called a permutation. Bijective functions are essential to many areas of including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group. Satisfying properties and means that a bijection is a function with domain X and it is more common to see properties and written as a single statement, Every element of X is paired with exactly one element of Y. Functions which satisfy property are said to be onto Y and are called surjections, Functions which satisfy property are said to be one-to-one functions and are called injections. With this terminology, a bijection is a function which is both a surjection and an injection, or using words, a bijection is a function which is both one-to-one and onto. Consider the batting line-up of a baseball or cricket team, the set X will be the players on the team and the set Y will be the positions in the batting order The pairing is given by which player is in what position in this order. Property is satisfied since each player is somewhere in the list, property is satisfied since no player bats in two positions in the order. Property says that for each position in the order, there is some player batting in that position, in a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them all to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. The instructor was able to conclude there were just as many seats as there were students. For any set X, the identity function 1X, X → X, the function f, R → R, f = 2x +1 is bijective, since for each y there is a unique x = /2 such that f = y. In more generality, any linear function over the reals, f, R → R, f = ax + b is a bijection, each real number y is obtained from the real number x = /a. The function f, R →, given by f = arctan is bijective since each real x is paired with exactly one angle y in the interval so that tan = x
23.
Restriction (mathematics)
–
In mathematics, the restriction of a function f is a new function f|A obtained by choosing a smaller domain A for the original function f. The notation f ↾ A is also used, let f, E → F be a function from a set E to a set F, so that the domain of f is in E. The restriction of the function f, R → R, x ↦ x 2 to R + = [0, ∞ ) is the injection f, R + → R, x ↦ x 2. The factorial function is the restriction of the function to the integers. Restricting a function f, X → Y to its entire domain X gives back the original function, i. e. f | X = f. Restricting a function twice is the same as restricting it once, i. e. if A ⊆ B ⊆ d o m f, the restriction of the identity function on a space X to a subset A of X is just the inclusion map of A into X. The restriction of a function is continuous. For a function to have an inverse, it must be one-to-one, if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function f = x 2 is not one-to-one, however, the function becomes one-to-one if we restrict to the domain x ≥0, in which case f −1 = y. The selection σ a θ v selects all those tuples in R for which θ holds between the a attribute and the value v. Thus, the selection operator restricts to a subset of the entire database. The pasting lemma is a result in topology that relates the continuity of a function with the continuity of its restrictions to subsets. Let X, Y be both closed subsets of a topological space A such that A = X ∪ Y, if f, A → B is continuous when restricted to both X and Y, then f is continuous. This result allows one to take two continuous functions defined on closed subsets of a space and create a new one. Sheaves provide a way of generalizing restrictions to objects besides functions, in sheaf theory, one assigns an object F in a category to each open set U of a topological space, and requires that the objects satisfy certain conditions. The most important condition is that there are restriction morphisms between every pair of objects associated to nested sets, i. e. If we have three open sets W ⊆ V ⊆ U, then the composite resW, V o resV, U = resW, the collection of all such objects is called a sheaf. If only the first two properties are satisfied, it is a pre-sheaf, more generally, the restriction A ◁ R of a binary relation R between E and F may be defined as a relation having domain A, codomain F and graph G = . Similarly, one can define a right-restriction or range restriction R ▷ B, indeed, one could define a restriction to n-ary relations, as well as to subsets understood as relations, such as ones of E × F for binary relations
24.
Inverse function
–
I. e. f = y if and only if g = x. As a simple example, consider the function of a real variable given by f = 5x −7. Thinking of this as a procedure, to reverse this and get x back from some output value, say y. In this case means that we should add 7 to y. In functional notation this inverse function would be given by, g = y +75, with y = 5x −7 we have that f = y and g = x. Not all functions have inverse functions, in order for a function f, X → Y to have an inverse, it must have the property that for every y in Y there must be one, and only one x in X so that f = y. This property ensures that a function g, Y → X will exist having the necessary relationship with f, let f be a function whose domain is the set X, and whose image is the set Y. Then f is invertible if there exists a g with domain Y and image X, with the property. If f is invertible, the g is unique, which means that there is exactly one function g satisfying this property. That function g is called the inverse of f, and is usually denoted as f −1. Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, not all functions have an inverse. For a function to have an inverse, each element y ∈ Y must correspond to no more than one x ∈ X, a function f with this property is called one-to-one or an injection. If f −1 is to be a function on Y, then each element y ∈ Y must correspond to some x ∈ X. Functions with this property are called surjections. This property is satisfied by definition if Y is the image of f, to be invertible a function must be both an injection and a surjection. If a function f is invertible, then both it and its inverse function f−1 are bijections, there is another convention used in the definition of functions. This can be referred to as the set-theoretic or graph definition using ordered pairs in which a codomain is never referred to, under this convention all functions are surjections, and so, being a bijection simply means being an injection. Authors using this convention may use the phrasing that a function is invertible if, the two conventions need not cause confusion as long as it is remembered that in this alternate convention the codomain of a function is always taken to be the range of the function. With this type of function it is impossible to deduce an input from its output, such a function is called non-injective or, in some applications, information-losing
25.
Partial function
–
In mathematics, a partial function from X to Y is a function f, X ′ → Y, for some subset X ′ of X. It generalizes the concept of an f, X → Y by not forcing f to map every element of X to an element of Y. If X ′ = X, then f is called a function and is equivalent to a function. Partial functions are used when the exact domain, X, is not known. Specifically, we say that for any x ∈ X, either. For example, we can consider the square root function restricted to the g, Z → Z g = n. Thus g is defined for n that are perfect squares. So, g =5, but g is undefined, there are two distinct meanings in current mathematical usage for the notion of the domain of a partial function. Most mathematicians, including recursion theorists, use the domain of f for the set of all values x such that f is defined. But some, particularly category theorists, consider the domain of a function f, X → Y to be X. Similarly, the range can refer to either the codomain or the image of a function. Occasionally, a function with domain X and codomain Y is written as f, X ⇸ Y. A partial function is said to be injective or surjective when the function given by the restriction of the partial function to its domain of definition is. A partial function may be both injective and surjective, because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective. An injective partial function may be inverted to a partial function. Furthermore, a function which is injective may be inverted to an injective partial function. The notion of transformation can be generalized to functions as well. A partial transformation is a function f, A → B, total function is a synonym for function
26.
Multivalued function
–
In mathematics, a multivalued function is a left-total relation. In the strict sense, a well-defined function associates one, and only one, the term multivalued function is, therefore, a misnomer because functions are single-valued. Multivalued functions often arise as inverses of functions that are not injective, such functions do not have an inverse function, but they do have an inverse relation. The multivalued function corresponds to this inverse relation, every real number greater than zero has two real square roots. The square roots of 4 are in the set, the square root of 0 is 0. Each complex number except zero has two roots, three cube roots, and in general n nth roots. The complex logarithm function is multiple-valued, the values assumed by log for real numbers a and b are log a 2 + b 2 + i arg +2 π n i for all integers n. Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic and we have tan = tan = tan = tan = ⋯ =1. As a consequence, arctan is intuitively related to several values, π/4, 5π/4, −3π/4 and we can treat arctan as a single-valued function by restricting the domain of tan x to −π/2 < x < π/2 – a domain over which tan x is monotonically increasing. Thus, the range of arctan becomes −π/2 < y < π/2 and these values from a restricted domain are called principal values. The indefinite integral can be considered as a multivalued function, the indefinite integral of a function is the set of functions whose derivative is that function. The constant of integration follows from the fact that the derivative of a constant function is 0 and these are all examples of multivalued functions that come about from non-injective functions. Since the original functions do not preserve all the information of their inputs, often, the restriction of a multivalued function is a partial inverse of the original function. Multivalued functions of a variable have branch points. For example, for the nth root and logarithm functions,0 is a point, for the arctangent function. Using the branch points, these functions may be redefined to be single-valued functions, by restricting the range. A suitable interval may be found through use of a branch cut, as in the case with real functions, the restricted range may be called principal branch of the function. Set-valued analysis is the study of sets in the spirit of mathematical analysis, instead of considering collections of only points, set-valued analysis considers collections of sets
27.
Implicit function
–
In mathematics, an implicit equation is a relation of the form R =0, where R is a function of several variables. For example, the equation of the unit circle is x 2 + y 2 −1 =0. An implicit function is a function that is defined implicitly by an implicit equation, thus, an implicit function for y in the context of the unit circle is defined implicitly by x 2 + f 2 −1 =0. This implicit equation defines f as a function of x only if −1 ≤ x ≤1, the implicit function theorem provides conditions under which a relation defines an implicit function. A common type of function is an inverse function. If f is a function of x, then the function of f. This solution is x = f −1, intuitively, an inverse function is obtained from f by interchanging the roles of the dependent and independent variables. Stated another way, the function gives the solution for x of the equation R = y − f =0. Example The product log is a function giving the solution for x of the equation y − x ex =0. An algebraic function is a function satisfies a polynomial equation whose coefficients are themselves polynomials. Algebraic functions play an important role in analysis and algebraic geometry. A simple example of a function is given by the left side of the unit circle equation. Solving for y gives a solution, y = ±1 − x 2. But even without specifying this explicit solution, it is possible to refer to the solution of the unit circle equation. Nevertheless, one can refer to the implicit solution y = g involving the multi-valued implicit function g. Not every equation R =0 implies a graph of a single-valued function, another example is an implicit function given by x − C =0 where C is a cubic polynomial having a hump in its graph. Thus, for a function to be a true function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a function only after zooming in on some part of the x-axis
28.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
29.
Real number
–
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
30.
Concentration
–
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished, mass concentration, molar concentration, number concentration, the term concentration can be applied to any kind of chemical mixture, but most frequently it refers to solutes and solvents in solutions. The molar concentration has variants such as concentration and osmotic concentration. To concentrate a solution, one must add more solute, or reduce the amount of solvent, by contrast, to dilute a solution, one must add more solvent, or reduce the amount of solute. Unless two substances are fully miscible there exists a concentration at which no further solute will dissolve in a solution, at this point, the solution is said to be saturated. If additional solute is added to a solution, it will not dissolve, except in certain circumstances. Instead, phase separation will occur, leading to coexisting phases, the point of saturation depends on many variables such as ambient temperature and the precise chemical nature of the solvent and solute. Concentrations are often called levels, reflecting the mental schema of levels on the axis of a graph. There are four quantities that describe concentration, The mass concentration ρ i is defined as the mass of a constituent m i divided by the volume of the mixture V, ρ i = m i V. The molar concentration c i is defined as the amount of a constituent n i divided by the volume of the mixture V, c i = n i V, however, more commonly the unit mol/L is used. The number concentration C i is defined as the number of entities of a constituent N i in a divided by the volume of the mixture V, C i = N i V. The volume concentration ϕ i is defined as the volume of a constituent V i divided by the volume of the mixture V, ϕ i = V i V. Being dimensionless, it is expressed as a number, e. g.0.18 or 18%, several other quantities can be used to describe the composition of a mixture. Note that these should not be called concentrations, normality is defined as the molar concentration c i divided by an equivalence factor f e q. Since the definition of the equivalence factor depends on context, IUPAC, the SI unit for molality is mol/kg. The mole fraction x i is defined as the amount of a constituent n i divided by the amount of all constituents in a mixture n t o t, x i = n i n t o t. However, the deprecated parts-per notation is used to describe small mole fractions. The mole ratio r i is defined as the amount of a constituent n i divided by the amount of all other constituents in a mixture
31.
Associative property
–
In mathematics, the associative property is a property of some binary operations. In propositional logic, associativity is a rule of replacement for expressions in logical proofs. That is, rearranging the parentheses in such an expression will not change its value, consider the following equations, +4 =2 + =92 × = ×4 =24. Even though the parentheses were rearranged on each line, the values of the expressions were not altered, since this holds true when performing addition and multiplication on any real numbers, it can be said that addition and multiplication of real numbers are associative operations. Associativity is not to be confused with commutativity, which addresses whether or not the order of two operands changes the result. For example, the order doesnt matter in the multiplication of numbers, that is. Associative operations are abundant in mathematics, in fact, many algebraic structures explicitly require their binary operations to be associative, however, many important and interesting operations are non-associative, some examples include subtraction, exponentiation and the vector cross product. Z = x = xyz for all x, y, z in S, the associative law can also be expressed in functional notation thus, f = f. If a binary operation is associative, repeated application of the produces the same result regardless how valid pairs of parenthesis are inserted in the expression. This is called the generalized associative law, thus the product can be written unambiguously as abcd. As the number of elements increases, the number of ways to insert parentheses grows quickly. Some examples of associative operations include the following, the two methods produce the same result, string concatenation is associative. In arithmetic, addition and multiplication of numbers are associative, i. e. + z = x + = x + y + z z = x = x y z } for all x, y, z ∈ R. x, y, z\in \mathbb. }Because of associativity. Addition and multiplication of numbers and quaternions are associative. Addition of octonions is also associative, but multiplication of octonions is non-associative, the greatest common divisor and least common multiple functions act associatively. Gcd = gcd = gcd lcm = lcm = lcm } for all x, y, z ∈ Z. x, y, z\in \mathbb. }Taking the intersection or the union of sets, ∩ C = A ∩ = A ∩ B ∩ C ∪ C = A ∪ = A ∪ B ∪ C } for all sets A, B, C. Slightly more generally, given four sets M, N, P and Q, with h, M to N, g, N to P, in short, composition of maps is always associative. Consider a set with three elements, A, B, and C, thus, for example, A=C = A
32.
Subset
–
In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T
33.
Absolute value
–
In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a x, |x| = −x for a negative x. For example, the value of 3 is 3. The absolute value of a number may be thought of as its distance from zero, generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, a value is also defined for the complex numbers. The absolute value is related to the notions of magnitude, distance. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English, the notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude, in programming languages and computational software packages, the absolute value of x is generally represented by abs, or a similar expression. Thus, care must be taken to interpret vertical bars as an absolute value sign only when the argument is an object for which the notion of an absolute value is defined. For any real number x the value or modulus of x is denoted by |x| and is defined as | x | = { x, if x ≥0 − x. As can be seen from the definition, the absolute value of x is always either positive or zero. Indeed, the notion of a distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Since the square root notation without sign represents the square root. This identity is used as a definition of absolute value of real numbers. The absolute value has the four fundamental properties, The properties given by equations - are readily apparent from the definition. To see that equation holds, choose ε from so that ε ≥0, some additional useful properties are given below. These properties are either implied by or equivalent to the properties given by equations -, for example, Absolute value is used to define the absolute difference, the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value cannot be directly generalised for a complex number
34.
Cubic function
–
In algebra, a cubic function is a function of the form f = a x 3 + b x 2 + c x + d, where a is nonzero. Setting f =0 produces an equation of the form. The solutions of this equation are called roots of the polynomial f, If all of the coefficients a, b, c, and d of the cubic equation are real numbers then there will be at least one real root. All of the roots of the equation can be found algebraically. The roots can also be found trigonometrically, alternatively, numerical approximations of the roots can be found using root-finding algorithms like Newtons method. The coefficients do not need to be complex numbers, much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3. The solutions of the cubic equation do not necessarily belong to the field as the coefficients. For example, some cubic equations with rational coefficients have roots that are complex numbers. Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, the problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, in the 3rd century, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations. In the 11th century, the Persian poet-mathematician, Omar Khayyám, in an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution, in the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of an equation, x3 + 12x = 6x2 +35. He used what would later be known as the Ruffini-Horner method to approximate the root of a cubic equation. He also developed the concepts of a function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the equation to find algebraic solutions to certain types of cubic equations. Leonardo de Pisa, also known as Fibonacci, was able to approximate the positive solution to the cubic equation x3 + 2x2 + 10x =20
35.
Commutative property
–
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says 3 +4 =4 +3 or 2 ×5 =5 ×2, the property can also be used in more advanced settings. The name is needed there are operations, such as division and subtraction. The commutative property is a property associated with binary operations and functions. If the commutative property holds for a pair of elements under a binary operation then the two elements are said to commute under that operation. The term commutative is used in several related senses, putting on socks resembles a commutative operation since which sock is put on first is unimportant. Either way, the result, is the same, in contrast, putting on underwear and trousers is not commutative. The commutativity of addition is observed when paying for an item with cash, regardless of the order the bills are handed over in, they always give the same total. The multiplication of numbers is commutative, since y z = z y for all y, z ∈ R For example,3 ×5 =5 ×3. Some binary truth functions are also commutative, since the tables for the functions are the same when one changes the order of the operands. For example, the logical biconditional function p ↔ q is equivalent to q ↔ p and this function is also written as p IFF q, or as p ≡ q, or as Epq. Further examples of binary operations include addition and multiplication of complex numbers, addition and scalar multiplication of vectors. Concatenation, the act of joining character strings together, is a noncommutative operation, rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order. The twists of the Rubiks Cube are noncommutative and this can be studied using group theory. Some non-commutative binary operations, Records of the use of the commutative property go back to ancient times. The Egyptians used the property of multiplication to simplify computing products. Euclid is known to have assumed the property of multiplication in his book Elements
36.
One-to-one function
–
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness, it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence, occasionally, an injective function from X to Y is denoted f, X ↣ Y, using an arrow with a barbed tail. A function f that is not injective is sometimes called many-to-one, however, the injective terminology is also sometimes used to mean single-valued, i. e. each argument is mapped to at most one value. A monomorphism is a generalization of a function in category theory. Let f be a function whose domain is a set X, the function f is said to be injective provided that for all a and b in X, whenever f = f, then a = b, that is, f = f implies a = b. Equivalently, if a ≠ b, then f ≠ f, in particular the identity function X → X is always injective. If the domain X = ∅ or X has only one element, the function f, R → R defined by f = 2x +1 is injective. The function g, R → R defined by g = x2 is not injective, however, if g is redefined so that its domain is the non-negative real numbers [0, +∞), then g is injective. The exponential function exp, R → R defined by exp = ex is injective, the natural logarithm function ln, → R defined by x ↦ ln x is injective. The function g, R → R defined by g = xn − x is not injective, since, for example, g = g =0. More generally, when X and Y are both the real line R, then a function f, R → R is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the line test. Functions with left inverses are always injections and that is, given f, X → Y, if there is a function g, Y → X such that, for every x ∈ X g = x then f is injective. In this case, g is called a retraction of f, conversely, f is called a section of g. Conversely, every injection f with non-empty domain has an inverse g. Note that g may not be an inverse of f because the composition in the other order, f o g. In other words, a function that can be undone or reversed, injections are reversible but not always invertible
37.
Derivative
–
The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation, the reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration, differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small