Operation (mathematics)
A function combining elements of a set into another element.
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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.
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. 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.
- arity
- 0, 1, 2, … (finite, but infinitary operations are sometimes considered)
- common types
- unary and binary
- classical operations
- addition, subtraction, multiplication, division
- examples of external operations
- scalar multiplication, inner product
- examples of non-numeric operations
- logic operations, set operations, function composition
Lore & Background
An operation is defined as a function from a Cartesian power of a set to the set itself. The number of operands is called the arity, which is usually a finite non-negative integer. A nullary operation (arity 0) is a constant. A partial operation is defined similarly but uses a partial mapping instead of a function. Operations may not be defined for every possible value of their domain; for example, division by zero is undefined in the real numbers.
Reader's Guide
Operations are fundamental to mathematics, providing the basic building blocks for arithmetic, algebra, and beyond. The four classical operations—addition, subtraction, multiplication, and division—underpin most calculations. Operations can be internal (e.g., vector addition) or external (e.g., scalar multiplication). They may have properties such as associativity, commutativity, or idempotence. The concept extends to logical values, sets, functions, and other mathematical objects. Partial operations, introduced by Bernard Neumann, allow for relations that are not total functions, expanding the study of universal algebra. Understanding operations is crucial for defining algebraic structures like groups, rings, and fields.
Did You Know?
- The mixed product is an example of a ternary operation (arity 3).
- An operation of arity 0, or nullary operation, is a constant.
- Operations can involve dissimilar objects, such as scalar multiplication of a vector by a scalar.
- A partial operation is defined similarly to an operation but with a partial mapping in place of a function.
Frequently Asked Questions
What is an Operation in mathematics?
An operation is a function that accepts a fixed number of elements from a set and returns an element of that same set. The number of inputs it requires is called its arity, and the inputs themselves go by the names operands or arguments.
What types of operations exist?
The most common are unary (one operand, such as additive inverse) and binary (two operands, such as addition or multiplication). Arity-0 operations simply serve as constants, and ternary operations like the mixed product take three inputs; infinitary operations are occasionally discussed as well.
What are the four classical operations?
They are addition, subtraction, multiplication, and division, which together form the backbone of arithmetic. These four are so foundational that they underpin virtually all of calculus and algebra.
How does an operation differ from a general function?
A general function can map between any two sets, whereas an operation specifically maps a set (or a product of copies of itself) back into that same set. This closed-domain-and-codomain requirement is what sets an operation apart from an arbitrary mapping.
Can operations be non-numeric?
Absolutely—logic gates, set-theoretic operations like union and intersection, and function composition all qualify as operations in this sense. External operations such as scalar multiplication and the inner product also appear, pulling in elements from a different algebraic structure.
More in Set Theory & Logic 1-19
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