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 valuestrue 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 of its domain. 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 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. 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, and the inner product operation on two vectors produces a quantity that is 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 "the addition operation", when focusing on the operands and result, but one switches to "addition operator", when focusing on the process, or from the more abstract viewpoint, the function.
Definition
An n-ary operationω from to Y is a function. The set is called the domain of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integern 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 -ary relation that is total on its n input domains and unique on its output domain. An n-ary partial operation ω from to Y is a partial function. An n-ary partial operation can also be viewed as an -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. There are obvious extensions where the arity is taken to be an infinite ordinal or cardinal, 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, 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 is called an internal operation. An n-ary operation where is called an external operation by the scalar set or operator setS. In particular for a binary operation, is called a left-external operation by S, and 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.
