Tuple
of these articles concerns itself with the mathematical term of the n - Tupels. For the term of the Tupels in computer science see tuple (computer science). |
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n - tuple in mathematics
the n - tuple is a term of mathematics. It designates an arranged composition ofObjects, contrary to quantities, whose elements do not have a fixed order.n - Tuple by round clips are usually indicated (here for <math> n=3< /math>, there three elements in tuple are present):
- <math> (A, b, C)< /math>
The objects become as elements, components or entries of the n - Tupels designates. Becausea n - Tuple everyone of its elements a clear place is assigned, can it also several times the same element contain. n designates here the number of elements of the n - Tupels. This number must be finite. In the case of a 2-Tupels one speaks also of an arranged pair, for n = 3 of a Tripel. The appropriate, rarely used words Quadrupel, Quintupel etc. gave - tuple the name to the n. Often the elements of a n become - Tupels with the help of the natural numbers indicates.
Some authors speak also of “tuple” (without n); Albrecht Beutelspacher guesses that in its mathematical style councellor “is o.B.d.A. trivially! “ from this linguistic carelessness.
Notation conflict: Often also open intervals are written as (A, b). Whether an interval or a pair is meant, is to be seen from the context.
demarcation in relation to quantities
a n - tuple is to be differentiated from a quantity to. With a quantity the sequence of the elements is insignificant. Therefore a quantity can never contain the same element several times. It can only contain it either, or it do not contain.
Forthe quantity stand curved clips, which mark that the elements unordered, i.e. without order, are.
(It is possible to extend the quantity term in such a way that an element can occur “several times”. See in addition to multi-quantity.)
examples
is A ≠ b, then is thatFew (A, b) differently of (b, A), on the other hand the quantity is {A, b} the same quantity as {b, A}.
“(90, 60, 90)” is a 3-Tupel and/or. a Tripel.
n - Tuple of numbers (or othershomogeneous objects) one calls vectors depending upon context also, as e.g. Elements of R^{ 3} or general R^{ n}. According to whether one it horizontal <math> (A, b, C)< /math> or vertically <math> \ begin {pmatrix} A \ \ b \ \ C \ end {pmatrix}< to /math> , speaks one writes of line or column vectors.
One notes however that those linear algebra a abstrakteren vector term uses: Vectors are defined as elements of a vector space. R n^{ (} with the obvious structure) is a vector space, but generally vectors are not in the sense of linear algebra a n - tuple. Vector in the sense of linear algebra tooits also no characteristic of an individual object, but only meaningful for an object as part of a whole with algebraic auxiliary structure, is a vector space.
applications
the term n - tuple is finally used with the definition of the cartesian product many quantities. In furtherConsequence is then needed the term of the cartesian product with the definition of the terms relation , function and consequence; n - Tuple is therefore a very fundamental term of mathematics.
formal definition
for n - tuple is above all demanded that two n - tuplethen and are alike only if they agree in all appropriate components:
<math> (a_1, a_2, \ dots, a_n) = (b_1, b_2, \ dots, b_n) \ quad \ Leftrightarrow \ quad a_1 = b_1 \, \, \ mathrm {and} \, \, a_2 = b_2 \, \, \ mathrm {and} \, \, \ dots \, a_n = b_n< /math>.
Several definitions of such n - tuple are usual:
definition of the n-Tupels with emphasis of the numbering
one understands a n - tuple with entries from a quantity of X (thus a n - digit “vector” with entries from X) as function of the quantity {1,…, n} into the quantity of X . A n- Tuple is thus a finite consequence, i.e. a function of a finite subset of the natural numbers.
With this definition one needs the term of the function for the condition. For this one e.g. becomes. pairs as, arranged first, above indicated define, then two digit functions and relationsas quantities of arranged pairs, in the next step n - tuple as special functions (consequences) and on that then n-digit functions and relations as quantities of n - tuples. This has the disadvantage that not only arranged pairs and 2-Tupel are different mathematical objects, but also two digit and n- digit relations (inclusive Functions) are differently structured.
inductive definition on the basis of the arranged pair
- an arranged pair <math> \ left (a_1, a_2 \ right)< /math> for example defined as the quantity <math> \ lbrace a_1, \ lbrace a_1, a_2 \ rbrace \ rbrace \! </math> or as the quantity <math> \ lbrace \ lbrace {a_1 \ rbrace}, \ lbrace a_1, a_2 \ rbrace \ rbrace \! </math>
- For <math> n > 2 </math> hasa n - Tuple the form <math> \ left (X, a_n \ right)< /math>, whereby <math> X< /math> (n − 1) - tuple is.
This definition becomes for example in (Lit.: Hlawka, binder, Schmitt) used and is the furthest common. It ensured not only the initially addressed principal claim at n - tuple, but permitsit also, the relation and Funktionsbegriff on the term of the n - to develop Tupels. It has however the disadvantage that with the formation of the n - Tupels did not intend number of digits when information is contained of tuple in. From kind of such defined n - tuple can (in contrast tofirst definition!) not to be opened whether it as 2-Tupel, 3-Tupel,…, (n − 1) - tuple or n - tuple to treat is, because it belongs to all these groups. Depending on, which group we assign it, e.g. falls. the result for the projection on thosesecond component of the n - Tupels differently out are actually applicable all components except first for this. An alteration of the definition, which avoids this disadvantage, is easily possible. For example one can for n > 2 a n - define tuple as a pair, which tothe number of places of n of the n - Tupels contains and on the other hand an arranged pair, whose first component (n − 1) - tuple is, thus:
- <math> (a_1, \ dots, a_n): = (((a_1, \ dots, a_ {n-1}), a_n), n)< /math>
inductive definition without use of the arranged pair
a definition of theN-Tupels without a condition of the term of the arranged pair is for example in (Lit.: Hazewinkel, 1993):
- <math> n=0: \ quad (): = \ lbrace \ rbrace< /math>
- <math> n=1: \ quad (A): = \ lbrace (), \ lbrace A \ rbrace \ rbrace< /math>
- <math> n> 1: \ quad (a_1, \; \ dots \; a_n): = \ lbrace (a_1, \; \ dots \; a_ {n-1}), \ lbrace a_n \ rbrace \ rbrace< /math>
According to this definition is the 2-Tupel <math> \; (A, b)< /math> the quantity <math> \; \ lbrace \ lbrace \ lbrace \ rbrace, \ lbrace A \ rbrace\ rbrace, \ lbrace b \ rbrace \ rbrace< /math>. On the other hand is the arranged pair <math> \; \ langle A, b \ rangle< /math> the quantity <math> \; \ lbrace \ lbrace A \ rbrace, \ lbrace A, b \ rbrace \ rbrace </math> or <math> \; \ lbrace A, \ lbrace A, b \ rbrace \ rbrace </math>, depending on, which definition of the arranged pair is at the basis put.
With this definition of the n-Tupels is thoseInformation about the number of components n-tuple contained, it applies thus the stricter characteristic
< math> (a_1, a_2, \ dots, a_n) = (b_1, b_2, \ dots, b_m) \ quad \ for Leftrightarrow \ quad m=n \, \, \ mathrm {and} \, \, a_1 = b_1 \, \, \ mathrm {and} \, \, a_2 = b_2 \, \, \ mathrm {and} \, \, \ dots \, a_n = b_n< /math>.
literature
- Edmund Hlawka, Christa binder, Peter Schmitt:Fundamental ideas of mathematics. Prugg publishing house, Vienna 1979. ISBN 3-85385-038-3.
- Hazewinkel, Michiel (OD.), Encyclopaedia OF Mathematics, volume 9, stochastics approximation - Zygmund class OF of transmitting ion. Kluwer Academic Publishers 1993. ISBN 1-55608-010-7.
See also: Set theory, complex number, Vector