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Operation (mathematics)

 

 
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Operation (mathematics)
 
 
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In its simplest meaning in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 and logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary
Unary operation

In mathematics, a unary operation is an operation with only one operand, i.e. an operation with a single input, or in other words, a function of one variable ....
 and binary
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
. Unary operations involve only one value, such as negation
Negation

In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true....
 and trigonometric function
Trigonometric function

In mathematics, the trigonometric functions are function s of an angle. They are important in the trigonometry of Triangle and modeling Periodic function, among many other applications....
s. Binary operations, on the other hand, take two values, and include addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
, subtraction
Subtraction

Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with....
, multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
, division
Division (mathematics)

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.Specifically, if c times b equals a, written:...
, and exponentiation
Exponentiation

Exponentiation is a mathematics operation , written 'an', involving two numbers, the base a and the exponent n....
.

Operations can involve mathematical objects other than numbers.






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In its simplest meaning in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 and logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary
Unary operation

In mathematics, a unary operation is an operation with only one operand, i.e. an operation with a single input, or in other words, a function of one variable ....
 and binary
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
. Unary operations involve only one value, such as negation
Negation

In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true....
 and trigonometric function
Trigonometric function

In mathematics, the trigonometric functions are function s of an angle. They are important in the trigonometry of Triangle and modeling Periodic function, among many other applications....
s. Binary operations, on the other hand, take two values, and include addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
, subtraction
Subtraction

Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with....
, multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
, division
Division (mathematics)

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.Specifically, if c times b equals a, written:...
, and exponentiation
Exponentiation

Exponentiation is a mathematics operation , written 'an', involving two numbers, the base a and the exponent n....
.

Operations can involve mathematical objects other than numbers. The logical values true and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted. Rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
s can be combined using the function composition
Function composition

In mathematics, a composite function represents the application of one function to the results of another. For instance, the functions and can be composed by first computing a f and then applying a function g to the output of f....
 operation, performing the first rotation and then the second. Operations on sets include the binary operations union and intersection and the unary operation of complementation. Operations on function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
s include composition
Function composition

In mathematics, a composite function represents the application of one function to the results of another. For instance, the functions and can be composed by first computing a f and then applying a function g to the output of f....
 and convolution
Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operator on two function s f and g, producing a third function that is typically viewed as a modified version of one of the original functions....
.

Operations may not be defined for every possible value. For example, in the real numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined form a set called its domain
Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
. The set which contains the values produced is called the codomain
Codomain

In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
, but the set of actual values attained by the operation is its range
Range (mathematics)

In mathematics, the range of a function is the Set of all "output" values produced by that function. Sometimes it is called the , or more precisely, the image of the domain of the function....
. For example, in the real numbers, the squaring operation only produces nonnegative numbers; the codomain is the set of real numbers but the range is the nonnegative numbers.

Operations can involve dissimilar objects. A vector can be multiplied by a scalar
Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
 to form another vector. And the inner product operation on two vectors produces a scalar. An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, and so on.

The values combined are called operands, arguments, or inputs, and the value produced is called the value, result, or output. Operations can have fewer or more than two inputs.

An operation is like an operator
Operator

In mathematics, an operator is a function which operates on another function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which ar...
, but the point of view is different. For instance, one often speaks of "the operation of addition" or "addition operation" when focusing on the operands and result, but one says "addition operator" (rarely "operator of addition") when focusing on the process, or from the more abstract viewpoint, the function +: S×S ? S.

General definition

An operation ? is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 of the form ? : X1 × … × Xk ? Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k (the number of arguments) is called the type or arity
Arity

In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the number of domains in the corresponding Cartesian product....
 of the operation. Thus a unary operation
Unary operation

In mathematics, a unary operation is an operation with only one operand, i.e. an operation with a single input, or in other words, a function of one variable ....
 has arity one, and a binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
 has arity two. An operation of arity zero, called a nullary operation, is simply an element of the codomain Y. An operation of arity k is called a k-ary operation. Thus a k-ary operation is a (k+1)-ary relation
Relation (mathematics)

In mathematics , a relation is a property that assigns truth values to combinations of k first-order logic. Typically, the property describes a possible connection between the components of a k-tuple....
 that is functional on its first k domains.

The above describes what is usually called a finitary operation, referring to the finite number of arguments (the value k). There are obvious extensions where the arity is taken to be an infinite ordinal or cardinal
Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of Set ....
, or even an arbitrary set indexing the arguments.

Often, use of the term operation implies that the domain of the function is a power of the codomain, although this is by no means universal (see the examples above).

See also

  • Algebra
    Algebra

    Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....


Special cases

  • Unary operation
    Unary operation

    In mathematics, a unary operation is an operation with only one operand, i.e. an operation with a single input, or in other words, a function of one variable ....
  • Binary operation
    Binary operation

    In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....


Related topics

  • Arity
    Arity

    In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the number of domains in the corresponding Cartesian product....
  • Binary relation
    Binary relation

    In mathematics, a binary relation is an arbitrary association of elements within a set or with elements of another set.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a divisibility of p, and no othe...
  • Domain
    Domain (mathematics)

    In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
  • Function
    Function (mathematics)

    The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
  • Multigrade operator
    Multigrade operator

    In logic and mathematics, a multigrade operator is a parametric operator with parameter k in the set N of non-negative integers....
  • Operator
    Operator

    In mathematics, an operator is a function which operates on another function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which ar...
  • Parametric operator
    Parametric operator

    In logic and mathematics, a parametric operator with parameter in the parametric set is a family of operators with index in the index set ....
  • Relation
    Relation (mathematics)

    In mathematics , a relation is a property that assigns truth values to combinations of k first-order logic. Typically, the property describes a possible connection between the components of a k-tuple....
  • Triadic relation
    Triadic relation

    In logic and mathematics, a triadic relation or a ternary relation is an important special case of a relation , one in which the number of places in the relation is three....