Operation (mathematics)

Source: Wikipedia, the free encyclopedia.
Not to be confused with Operator (mathematics).
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.

The most commonly studied operations are

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 and , and returns the result .

There are two common types of operations:

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

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

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 operation ω 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 operation ω 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

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 multioperation ω 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

References

  1. ^ a b c d "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. 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.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1216682116"