Binary operation

In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities. Binary operations are sometimes called dyadic operations in order to avoid confusion with the binary numeral system. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division.

Related Topics:
Mathematics - Binary - Operator - Binary numeral system - Arithmetic - Addition - Subtraction - Multiplication - Division

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More precisely, a binary operation on a set S is a binary function from S and S to S, in other words a function f from the Cartesian product S × S to S.

Related Topics:
Set - Binary function - Cartesian product

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Sometimes, especially in computer science, the term is used for any binary function. That f takes values in the same set S that provides its arguments is the property of closure.

Related Topics:
Computer science - Binary function - Closure

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Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more.

Related Topics:
Abstract algebra - Groups - Monoid - Semigroup - Ring

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Most generally, a magma is a set together with any binary operation defined on it.

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Many binary operations of interest in both algebra and formal logic are commutative or associative.

Related Topics:
Commutative - Associative

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Many also have identity elements and inverse elements.

Related Topics:
Identity element - Inverse element

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Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions on a single set.

Related Topics:
Addition - Multiplication - Number - Matrices - Composition of functions

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Examples of operations that are not commutative are subtraction (-), division (/), exponentiation(^), and super-exponentiation(@).

Related Topics:
Commutative - Subtraction - Division - Exponentiation - Super-exponentiation

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Binary operations are often written using infix notation such as a * b, a + b, or a · b rather than by functional notation of the form f(a,b).

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Sometimes they are even written just by juxtaposition: ab.

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They can also be expressed using prefix or postfix notations. A prefix notation, Polish notation, dispenses with parentheses; it is probably more often encountered now in its postfix form, reverse Polish notation.

Related Topics:
Polish notation - Reverse Polish notation

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