Linkage (math)

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Illustration of a two-digit link that returns the result from the two arguments and . ${\displaystyle \circ }$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x\circ y}$

In mathematics , linkage is used as a generic term for various operations : In addition to the basic arithmetic arithmetic operations ( addition , subtraction , etc.), it also covers geometric operations (such as mirroring , rotation, etc.) and other arithmetic operations and occasionally logical operators . A link defines how mathematical objects of the same or similar type determine another object with one another. In the case of a relatively small number of elements and a link with only a few, such as two places, where elements can be used as operands, this definition is clearly possible using a link table in which, for example, B. for a 2-digit link all possible pairings are listed and their result is given, the result of the calculation.

The word linkage is also used to denote the sequential execution (concatenation) of functions .

general definition

For a natural number, let sets and another set be given. Then every mapping of the Cartesian product is referred to as a -digit link . Such a link uniquely assigns an element of the set to each tuple with . Of course, the quantities and can partly or wholly coincide. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle A_{1},\dotsc ,A_{n}}$ ${\displaystyle B}$ ${\displaystyle A_{1}\times \dotsb \times A_{n}}$ ${\displaystyle B}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle (x_{1},\dotsc ,x_{n})}$ ${\displaystyle x_{1}\in A_{1},\;\dotsc ,\;x_{n}\in A_{n}}$ ${\displaystyle B}$ ${\displaystyle A_{1},\dotsc ,A_{n}}$ ${\displaystyle B}$

In the special case that only occurs, the link becomes ${\displaystyle B}$ ${\displaystyle A_{i}=B\ \mathrm {f{\ddot {u}}r} \ 1\leq i\leq n,}$

${\displaystyle \underbrace {B\times \dotsb \times B} _{n{\text{-mal}}}\to B}$

inner -ary link ${\displaystyle n}$ or -ary operation on called. Occurs at least once among them, for example ${\displaystyle n}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle A_{i}}$

${\displaystyle A_{i}\neq B\ \mathrm {f{\ddot {u}}r} \ 1\leq i\leq m}$ and ${\displaystyle A_{i}=B\ \mathrm {f{\ddot {u}}r} \ m+1\leq i\leq n}$

for a with so the link is called the outer -digit link on with the operator area . The elements of are then called operators. ${\displaystyle m}$ ${\displaystyle 0\leq m<n,}$ ${\displaystyle n}$ ${\displaystyle B}$ ${\displaystyle A_{1}\times \dotsb \times A_{m}}$ ${\displaystyle A_{1}\times \dotsb \times A_{m}}$

An inner -digit link on can also be viewed as an outer two-digit link on with the operator area. ${\displaystyle n}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle B^{n-1}}$

Every -digit link can be understood as a -digit relation . ${\displaystyle n}$ ${\displaystyle (n+1)}$

Examples

• By

${\displaystyle (x,y,z)\mapsto {\frac {x+y}{z^{2}+1}}}$

defined mapping from to is a three-digit link or inner three-digit link to . ${\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

• If a picture is from to , then it is through ${\displaystyle f}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

${\displaystyle *\colon \,\{f\}\times \mathbb {R} \to \mathbb {R} ,\,(f,x)\mapsto f*x:=f(x)}$

(each of the image and an element of the formed pair of the image of this element is below the figure assigned) ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle R}$ ${\displaystyle f}$

given an outer two-digit link on with operator scope and the only operator . ${\displaystyle \mathbb {R} }$ ${\displaystyle \{f\}}$ ${\displaystyle f}$

Zero-digit links

As a zero-digit combination of a lot for a lot one can figure out of to be considered. It applies ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A^{0}}$ ${\displaystyle B}$

${\displaystyle A^{0}=A^{\emptyset }=\{f\mid f\colon \,\emptyset \to A\}=\{\emptyset \}=\{0\}=1,}$

therefore, each of these figures can be specified as follows:

${\displaystyle \operatorname {c} _{b}\colon \{\emptyset \}\to B,\,\emptyset \mapsto b,}$ for a ${\displaystyle b\in B.}$

Each zero-digit link is therefore constant and can in turn be understood as the constant . ${\displaystyle \operatorname {c} _{b}\in B^{\{\emptyset \}}=B^{1}}$ ${\displaystyle b\in B}$

Since it always applies, every zero-digit link can be viewed as an inner link on : ${\displaystyle B^{0}=\{\emptyset \}}$ ${\displaystyle \{\emptyset \}\to B}$ ${\displaystyle B}$ ${\displaystyle B^{0}\to B.}$

One-digit links

→ Main article : One-digit shortcut

Single-digit links are mappings of a set after a set . ${\displaystyle A}$ ${\displaystyle B}$

Examples

• A lot is given . For each element of the power set , i.e. for each subset of , let us define: ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle P(A)}$ ${\displaystyle X}$ ${\displaystyle A}$

${\displaystyle {}^{\operatorname {c} }\colon X\mapsto X^{\operatorname {c} }:=A\setminus X}$( Complement of ${\displaystyle X}$ ).

• The sine function

${\displaystyle \sin \colon \mathbb {R} \to \mathbb {R} ,x\mapsto \sin(x),}$

is a one-digit shortcut.

Two-digit (binary) links

→ Main article : Two-digit link

The term “link” is particularly often used in the sense of a two-digit link. Important special cases are internal and external connections. Two-digit links are often noted in infix notation, i.e. with a symbol such as a plus sign between the two operands.

Three- and multi-digit links

Rarely one speaks of three- and multi-digit links . Examples of a three-digit link are:

• the mapping, which each assigns three vectors from the their late product (from ) and ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} }$
• the ternary link in a ternary body .

Partial links

If in the above definition for (total) links the term of the (totally understood) mapping is replaced by partial mapping , then one speaks of a partial link : It is then permitted that a link value ( ie image value, function value) is assigned.

Links in algebra

In algebra , links are used to define algebraic structures . The links must meet certain conditions ( axioms ). In the case of partial algebras , partial links are also permitted.

For example, a semigroup is a set with an interior two-digit link that satisfies the associative law. The requirement that the result of the connection should again be an element of the given set (isolation) is already contained in the definition of the inner connection.

Web links

Wikibooks: Math for Non-Freaks: Linking  - Learning and Teaching Materials