The general operation as explained on this page should not be confused with the more
specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic
operation. In its simplest meaning in mathematics and
logic, an operation is an action or procedure which produces a new value from one or more
input values, called "operands". There are two common types of operations: unary and
binary. Unary operations involve only one value, such as negation and trigonometric
functions. Binary operations, on the other hand, take two values, and include addition,
subtraction, multiplication, division, and exponentiation.
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. 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. Operations
on functions include composition and convolution. 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. The set which contains the values produced is called the codomain, but the set
of actual values attained by the operation is its range. 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 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, 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" when focusing on the process, or
from the more abstract viewpoint, the function +: S×S → S.
General description An operation ω is a function of the form
ω : V → Y, where V ⊂ X1 × … × Xk. The sets Xk are called the domains of the
operation, the set Y is called the codomain of the operation, and the fixed non-negative
integer k is called the type or 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 operation of arity k is called a k-ary
operation. Thus a k-ary operation is a-ary relation 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. There are obvious extensions
where the arity is taken to be an infinite ordinal or cardinal, 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,
as in the example of multiplying a vector by a scalar.
See also Algebra
Unicode mathematical operators Special cases
Unary operation Binary operation
Related topics Notes
^ See e.g. Chapter II, Definition 1.1 in: S. N. Burris and H. P. Sankappanavar, A Course
in Universal Algebra, Springer, 1981. [1]