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Information about Binary Operation

In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator. 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.

More precisely, a binary operation on a set is a binary function that maps elements of the Cartesian product to :
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.

Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more. Most generally, a magma is a set together with any binary operation defined on it.

Many binary operations of interest in both algebra and formal logic are commutative or associative. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions on a single set.

If is not a function, but is instead a partial function, it is called a partial operation. For instance, division of real numbers is a partial function, because one can't divide by zero: 1/0 and 0/0 are not defined, for instance.

An example of a operation that is not commutative is subtraction (-). Examples of partial operations that are not commutative include division (/), exponentiation(^), and super-exponentiation(@).

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). Sometimes they are even written just by : ab. Powers are usually also written without operator, but with the second argument as superscript.

Binary operations sometimes use prefix or postfix notation, this dispenses with parentheses. Prefix notation is also called Polish notation; postfix notation, also called reverse Polish notation, is probably more often encountered.

Pair and tuple

A binary operation, ab, depends on the ordered pair (a,b) and so (ab)c (where the parentheses here mean first operate on the ordered pair (a,b) and then operate on the result of that using the ordered pair ((ab),c)) depends in general on the ordered pair ((a,b),c). Thus, for the general, non-associative case, binary operations can be represented with binary trees.

However:
  • If the operation is associative, (ab)c=a(bc), then the value depends only on the tuple (a,b,c).
  • If the operation is commutative, ab=ba, then the value depends only on the multiset { {a,b},c}.
  • If the operation is both associative and commutative then the value depends only on the multiset {a,b,c}.
  • If the operation is both associative and commutative and idempotent, aa=a, then the value depends only on the set {a,b,c}.

External binary operations

An external binary operation is a binary function from K and S to S. This differs from a binary operation in the strict sense in that K need not be S; its elements come from outside.

An example of an external binary operation is scalar multiplication in linear algebra. Here K is a field and S is a vector space over that field.

An external binary operation may alternatively be viewed as an action; K is acting on S.

See also

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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In logic, mathematics, and computer science, the arity (synonyms include type, adicity, and rank) of a function or operation is the number of arguments or operands that the function takes.
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In mathematics, a binary function, or function of two variables, is a function which takes two inputs.

Precisely stated, a function is binary if there exists sets such that
Where is the Cartesian product of and

For example, if Z
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operator is a function, that operates on (or modifies) another function. Often, an "operator" is a function that acts on functions to produce other functions (the sense in which Oliver Heaviside used the term); or it may be a generalization of such a function, as in linear algebra,
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binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2.
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Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business
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Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection.
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Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. Subtraction is denoted by a minus sign in infix notation.

The traditional names for the parts of the formula
c − b = a
are
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Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:


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In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.

Specifically, if c times b equals a, written:
where b is not zero, then a
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SET may stand for:
  • Sanlih Entertainment Television, a television channel in Taiwan
  • Secure electronic transaction, a protocol used for credit card processing,

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In mathematics, a binary function, or function of two variables, is a function which takes two inputs.

Precisely stated, a function is binary if there exists sets such that
Where is the Cartesian product of and

For example, if Z
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In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept.
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Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems.
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In mathematics, a binary function, or function of two variables, is a function which takes two inputs.

Precisely stated, a function is binary if there exists sets such that
Where is the Cartesian product of and

For example, if Z
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In mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 7 are both natural numbers, but the result of 3
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Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Most authors nowadays simply write algebra instead of abstract algebra.
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group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory.
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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. Monoids occur in a number of branches of mathematics.
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In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. In other words, a semigroup is an associative magma. The terminology is derived from the anterior notion of a group.
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In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. The branch of abstract algebra which studies rings is called ring theory.
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In mathematics, particularly in abstract algebra, a magma (or groupoid) is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M × M → M.
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Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result. It is a fundamental property in most branches of mathematics and many proofs depend on it.
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associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed.
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identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. This is used for groups and related concepts.
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In mathematics, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element.
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Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection.
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Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:


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number is an abstract idea used in counting and measuring. A symbol which represents a number is called a numeral, but in common usage the word number is used for both the idea and the symbol.
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matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied.
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