Operation (mathematics)

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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) Arithmetic operations.svg
Elementary arithmetic operations:
  • +, plus (addition)
  • −, minus (subtraction)
  • ÷, obelus (division)
  • ×, times (multiplication)

In mathematics, an operation is a function from a set to itself. For example, an operation on real numbers will take in real numbers and return a real number. An operation can take 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] 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 (where X is a set such as the set of real numbers).

Definition

An n-ary operationω on a set X is a function ω: Xn → X. The set Xn is called the domain of the operation, the output set 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 Xn to X is a partial function ω: Xn → X. 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 . [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.

<span class="mw-page-title-main">Logical conjunction</span> Logical connective AND

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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.

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In mathematics, an algebraic structure consists of a nonempty set A, a collection of operations on A, and a finite set of identities 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 ⁠⁠, where A is a set. The function ⁠⁠ is a unary operation on A.

In mathematics, the composition operator takes two functions, and , and returns a new function . Thus, the function g is applied after applying f to x.

<span class="mw-page-title-main">Range of a function</span> Subset of a functions codomain

In mathematics, the range of a function may refer to either of two closely related concepts:

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.

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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 .
  4. ↑ Weisstein, Eric W. "Binary Operation". MathWorld .
  5. ↑ Weisstein, Eric W. "Vector". MathWorld . "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. "Coomain". MathWorld .
  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.
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