Binary operation

A binary operation, also called binary logic, is in mathematics a link that has exactly two operands. Two-digit shortcuts occur in particular in algebra very common and it is called there abbreviated by link without adding double digits. There are also links to other arity that link to each other, for example, three or more operands.

• 2.1 Examples

• 3.1 Examples
• 3.2 Remarks

• 4.1 Examples

Definition

A binary operation is a mapping from the Cartesian product of two sets, and after a third set. Such a link assigns each ordered pair (a, b ) of elements and as the two operands with an element to be the result or outcome of the link. If the amounts, and are the same, the link is also called inner join; otherwise it is called an outer join.

Spellings

Two-digit shortcuts to write often in infix notation instead of the usual prefix notation. For example, to write an addition as instead of. Multiplication is often written without any symbol, ie. The best known is the postfix reverse Polish notation, which does not require brackets. The chosen notation, whether prefix, infix, or postfix depends essentially on the usefulness in the given context and the respective traditions.

Examples

• The basic arithmetic operations (addition, multiplication, subtraction and division) on corresponding sets of numbers are two digit shortcuts. For example, results from the division of an integer by a natural number a rational number. This therefore corresponds to a link.

• The composition of mappings is a binary operation: It assigns each picture and each picture their sequential execution to. This therefore corresponds to a link. Here, the quantities, and are chosen arbitrarily. This association occurs in almost all areas of mathematics and is based on category theory.

Inner binary operation

An inner two-digit or two-digit shortcut operation on a set is a binary operation, which therefore assigns to each ordered pair of an element of. This corresponds to the above general definition in the special case. The additional attribute inner expresses that all operands of the set and the link are not leading out. We say also, is closed with respect.

Inside two digit shortcuts are an important part of algebraic structures that are investigated in abstract algebra. They occur in semigroups, monoids, groups, rings, and other mathematical structures.

More generally called a lot with any inner join also magma. Often such links have additional properties, for example they are associative or commutative. Many also have a neutral element and invertible elements.

Examples

• The addition and multiplication of integers are inner joins or. The same is true for the natural, rational, real and complex numbers.
• The subtraction of integers is an inner join. The same applies for the rational real and complex numbers. Note, however, that the subtraction of natural numbers leading out of the set of natural numbers, and thus has no inner link. ( Here, for example ).
• The rational numbers without division is an inner join. The same applies to the real and complex numbers respectively without. Note, however, that the division of whole numbers leading out of the set of integers, and therefore is not an inner join. ( Here, for example ).
• For a given quantity are the averages and the union of subsets inner joins on the power set.
• For each set, the composition of mappings is an inner join.

Outer two digit shortcuts first kind

An outer two digit shortcuts first type is a binary operation which one is called right action of on, or links to operation calls from to. They differ from inner two digit shortcuts fact that the designated amount as operator domain whose elements are called operators, not necessarily a subset of is, that may come from outside. One says then operated from the right or from the left and the elements of hot right or left operators.

Through each operator is exactly one figure or defined, which is also called to the transformation. When multiplied to write instead or also briefly respectively, and it is usually distinguished between the operator and the associated transformation or not. It then writes to the so-called Operator notation: or

Examples

• For every natural number an inner - digit shortcut is always an external binary operation of the first kind, namely both a right-and a left action of on (it's always ). Such internal links are, therefore, also commonly referred to as digit operations. A zero digit combination can be interpreted as inner join and therefore always considered as zero digit operation.

• In a group operation is a group and a lot. It also requires a certain tolerance of this operation with the group structure namely and for all, and the neutral element of

• In linear algebra is in the scalar multiplication of the operator domain, a body, and most or an Abelian group, such as addition or Man Calls for a suitable compatibility of the scalar multiplication with the already existing structures and Equipped with surgery is a vector space

Remark

The term operation or operator, for example, in functional analysis, for general two-digit shortcuts or needed. These quantities are the same (usually algebraic ) structure, and often is the transformation or the structure and be compatible.

Outer two digit shortcuts second nature

An external binary operation of the second kind is a figure that is is a binary operation on a set but must be re not finished, so it may also apply.

Examples

• Each inner binary operation is also an external binary operation of the second kind

• The scalar product in the -dimensional vector space assigned to each of two vectors is a real number and is thus an external binary operation of the second kind for the scalar product and an inner binary operation, but not for.

• The scalar product in the skew field of quaternions is an internal binary operation and thus an external binary operation of the second kind Summing up, however, as four-dimensional division algebra over on, then the dot product is no internal link more, but it remains an external binary operation of the second kind

• Is an affine space over a vector space so is with an external binary operation of the second kind