Operation (mathematics)

Last updated
Elementary arithmetic operations: .mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0} +, plus (addition) -, minus (subtraction) /, obelus (division) x, times (multiplication) Basic arithmetic operators.svg
Elementary arithmetic operations:
  • +, plus (addition)
  • −, minus (subtraction)
  • ÷, obelus (division)
  • ×, times (multiplication)

In mathematics, an operation is a function which takes zero or more input values (also called " operands " or "arguments") to a well-defined output value. The number of operands is the arity of the operation.

Contents

The most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and multiplication, and unary operations (i.e., operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant. [1] [2] The mixed product is an example of an operation of arity 3, also called ternary operation.

Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, [1] in which case the "usual" operations of finite arity are called finitary operations.

A partial operation is defined similarly to an operation, but with a partial function in place of a function.

Types of operation

A binary operation takes two arguments x {\displaystyle x} and y {\displaystyle y} , and returns the result x [?] y {\displaystyle x\circ y} . Binary operations as black box.svg
A binary operation takes two arguments and , and returns the result .

There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. [3] Binary operations, on the other hand, take two values, and include addition, subtraction, multiplication, division, and exponentiation. [4]

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. [5] Rotations can be combined using the function composition 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 . [6] [7] [8] Operations on functions include composition and convolution. [9] [10]

Operations may not be defined for every possible value of its domain . For example, in the real numbers one cannot divide by zero [11] or take square roots of negative numbers. The values for which an operation is defined form a set called its domain of definition or active domain. The set which contains the values produced is called the codomain , but the set of actual values attained by the operation is its codomain of definition, active codomain, image or range . [12] [ failed verification ] For example, in the real numbers, the squaring operation only produces non-negative numbers; the codomain is the set of real numbers, but the range is the non-negative numbers.

Operations can involve dissimilar objects: a vector can be multiplied by a scalar to form another vector (an operation known as scalar multiplication), [13] and the inner product operation on two vectors produces a quantity that is scalar. [14] [15] 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 (including the case of zero input and infinitely many inputs [1] ).

An operator is similar to an operation in that it refers to the symbol or the process used to denote the operation, hence their point of view is different. For instance, one often speaks of "the operation of addition" or "the addition operation", when focusing on the operands and result, but one switches to "addition operator" (rarely "operator of addition"), when focusing on the process, or from the more symbolic viewpoint, the function +: X × X → X.

Definition

An n-ary operationω from X1, …, Xn to Y is a function ω: X1 × … × Xn → Y. The set X1 × … × Xn is called the domain of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer n (the number of operands) is called the arity of the operation. Thus a unary operation has arity one, and a binary operation has arity two. An operation of arity zero, called a nullary operation, is simply an element of the codomain Y. An n-ary operation can also be viewed as an (n + 1)-ary relation that is total on its n input domains and unique on its output domain.

An n-ary partial operationω from X1, …, Xn to Y is a partial function ω: X1 × … × Xn → Y. An n-ary partial operation can also be viewed as an (n + 1)-ary relation that is unique on its output domain.

The above describes what is usually called a finitary operation, referring to the finite number of operands (the value n). There are obvious extensions where the arity is taken to be an infinite ordinal or cardinal, [1] or even an arbitrary set indexing the operands.

Often, the use of the term operation implies that the domain of the function includes a power of the codomain (i.e. the Cartesian product of one or more copies of the codomain), [16] although this is by no means universal, as in the case of dot product, where vectors are multiplied and result in a scalar. An n-ary operation ω: Xn → X is called an internal operation. An n-ary operation ω: Xi × S × Xn − i − 1 → X where 0 ≤ i < n is called an external operation by the scalar set or operator setS. In particular for a binary operation, ω: S × X → X is called a left-external operation by S, and ω: X × S → X is called a right-external operation by S. An example of an internal operation is vector addition, where two vectors are added and result in a vector. An example of an external operation is scalar multiplication, where a vector is multiplied by a scalar and result in a vector.

An n-ary multifunction or multioperationω is a mapping from a Cartesian power of a set into the set of subsets of that set, formally ω: Xn → P(X). [17]

See also

Related Research Articles

In mathematics, a binary function is a function that takes two inputs.

<span class="mw-page-title-main">Binary operation</span> Mathematical operation with two operands

In mathematics, a binary operation or dyadic operation is a rule for combining two elements to produce another element. More formally, a binary operation is an operation of arity two.

In mathematics, a finitary relation over a sequence of sets X1, ..., Xn is a subset of the Cartesian product X1 × ... × Xn; that is, it is a set of n-tuples (x1, ..., xn), each being a sequence of elements xi in the corresponding Xi. Typically, the relation describes a possible connection between the elements of an n-tuple. For example, the relation "x is divisible by y and z" consists of the set of 3-tuples such that when substituted to x, y and z, respectively, make the sentence true.

In logic, mathematics, and computer science, arity is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency.

Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study.

In mathematics, an algebraic structure consists of a nonempty set A, a collection of operations on A, and a finite set of identities, known as axioms, that these operations must satisfy.

In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function f : A → A, where A is a set. The function f is a unary operation on A.

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.

In mathematics, function composition is an operation  ∘  that takes two functions f and g, and produces a function h = g  ∘  f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in domain X to g(f(x)) in codomain Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.

<span class="mw-page-title-main">Additive inverse</span> Number that, when added to the original number, yields zero

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. The operation taking a number to its additive inverse is known as sign change or negation. For a real number, it reverses its sign: the additive inverse of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself.

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.

In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: the input and output of a truth function are all truth values; a truth function will always output exactly one truth value, and inputting the same truth value(s) will always output the same truth value. The typical example is in propositional logic, wherein a compound statement is constructed using individual statements connected by logical connectives; if the truth value of the compound statement is entirely determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and any logical connectives used are said to be truth functional.

In mathematics, a unary function is a function that takes one argument. A unary operator belongs to a subset of unary functions, in that its codomain coincides with its domain. In contrast, a unary function's domain need not coincide with its range.

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value of some function An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise.

In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the multiplication operation to the product operation. Other operations, e.g., the set-theoretic operations union and intersection, are also often iterated, but the iterations are not given separate names. In print, summation and product are represented by special symbols; but other iterated operators often are denoted by larger variants of the symbol for the ordinary binary operator. Thus, the iterations of the four operations mentioned above are denoted

In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.

Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1. Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under some operations satisfying certain equations.

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic.

In mathematics and physics, vector is a term that refers informally to some quantities that cannot be expressed by a single number, or to elements of some vector spaces.

References

  1. 1 2 3 4 "Algebraic operation - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-10.
  2. ↑ DeMeo, William (August 26, 2010). "Universal Algebra Notes" (PDF). math.hawaii.edu. Archived from the original (PDF) on 2021-05-19. Retrieved 2019-12-09.
  3. ↑ Weisstein, Eric W. "Unary Operation". mathworld.wolfram.com. Retrieved 2020-07-27.
  4. ↑ Weisstein, Eric W. "Binary Operation". mathworld.wolfram.com. Retrieved 2020-07-27.
  5. ↑ Weisstein, Eric W. "Vector". mathworld.wolfram.com. Retrieved 2020-07-27. Vectors can be added together (vector addition), subtracted (vector subtraction) ...
  6. ↑ Weisstein, Eric W. "Union". mathworld.wolfram.com. Retrieved 2020-07-27.
  7. ↑ Weisstein, Eric W. "Intersection". mathworld.wolfram.com. Retrieved 2020-07-27.
  8. ↑ Weisstein, Eric W. "Complementation". mathworld.wolfram.com. Retrieved 2020-07-27.
  9. ↑ Weisstein, Eric W. "Composition". mathworld.wolfram.com. Retrieved 2020-07-27.
  10. ↑ Weisstein, Eric W. "Convolution". mathworld.wolfram.com. Retrieved 2020-07-27.
  11. ↑ Weisstein, Eric W. "Division by Zero". mathworld.wolfram.com. Retrieved 2020-07-27.
  12. ↑ Weisstein, Eric W. "Domain". mathworld.wolfram.com. Retrieved 2020-08-08.
  13. ↑ Weisstein, Eric W. "Scalar Multiplication". mathworld.wolfram.com. Retrieved 2020-07-27.
  14. ↑ Jain, P. K.; Ahmad, Khalil; Ahuja, Om P. (1995). Functional Analysis. New Age International. ISBN   978-81-224-0801-0.
  15. ↑ Weisstein, Eric W. "Inner Product". mathworld.wolfram.com. Retrieved 2020-07-27.
  16. ↑ Burris, S. N.; Sankappanavar, H. P. (1981). "Chapter II, Definition 1.1". A Course in Universal Algebra. Springer.
  17. ↑ Brunner, J.; Drescher, Th.; Pöschel, R.; Seidel, H. (Jan 1993). "Power algebras: clones and relations" (PDF). EIK (Elektronische Informationsverarbeitung und Kybernetik). 29: 293–302. Retrieved 2022-10-25.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.