In mathematics, an operation is a function that takes as input a fixed number of elements of a set and returns an element of the same set. For example, addition on real numbers is an operation that accepts two real numbers and returns a real number. In general, the input values may be called "operands" or "arguments". The number of operands is the arity of the operation. The arity is usually one of ${\displaystyle 0,1,2,\ldots }$.

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 0, or nullary operation, is a constant. The mixed product is an example of an operation of arity 3, also called ternary operation.

The four classical operations are addition, subtraction, multiplication, and division. These operations form the foundation of arithmetic and are essential for performing calculations and solving problems in various fields.

Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, 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.

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 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.