Operation (mathematics)

In mathematics, an operation is a function that takes a specific number of inputs from a set and produces an output from that same set. These inputs are called operands or arguments, and the count of operands is the operation's arity. The most familiar operations are binary (arity 2), like addition and multiplication, and unary (arity 1), like taking the additive inverse.

An operation with arity 0 is a constant. Ternary operations, such as the mixed product, also exist. The four classical operations—addition, subtraction, multiplication, and division—underpin arithmetic and are vital for calculations across many fields.

Definition

Arity is usually finite, but infinitary operations are sometimes considered, with finite-arity ones called finitary operations. A partial operation is similar but uses a partial mapping instead of a full function.

Types of operation

Operations can involve more than just numbers: logic operations combine truth values (and, or, not); vectors can be added or subtracted; rotations combine via composition; sets have union, intersection, and complement; and functions have composition and convolution. An operation may not be defined for all possible inputs—for instance, division by zero or square roots of negative numbers are undefined in the real numbers. The set of inputs for which it is defined is its domain of definition, while the set of actual outputs is its range or image.

Operations can combine dissimilar objects: scalar multiplication multiplies a vector by a scalar to produce a vector, while the inner product of two vectors yields a scalar. Operations may have properties like associativity, commutativity, or idempotence.

The symbol or process used to denote an operation is often called an operator. An n-ary operation on a set X is a function from Xⁿ to X, where n is the arity. A nullary operation is simply an element of X. An n-ary partial operation is a partial mapping from Xⁿ to X. These are typically finitary, but arity can be extended to infinite ordinals or cardinals.

An operation where the domain is a power of the codomain is called internal (e.g., vector addition). An operation that involves an external set S is called external—for example, left-external (S × X → X) or right-external (X × S → X), as in scalar multiplication. An n-ary multifunction maps from a Cartesian power to the power set of the codomain.