Restriction

In mathematics , the term constraint is mostly used to reduce the domain of a function .

It is also possible for relations to consider the restriction to a subset of the basic set.

Occasionally, in mathematical proofs, the phrase “ without restriction of generality ” (o. B. d. A.) is used. This has nothing to do with the mathematical terms explained here.

Restriction of a function

definition

If there is an arbitrary function and a subset of the definition set , then the restriction (or restriction ) of to is understood to be the function that corresponds to with . If the function is understood as a right-unambiguous, left-total relation, then this definition reproduces that of the pre-restriction . With the help of the inclusion map, the restriction can be briefly written as ${\displaystyle f\colon A\to B}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle f|_{X}}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle g\colon X\to B}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle \upharpoonleft }$ ${\displaystyle i\colon X\to A,\,x\mapsto x}$

${\displaystyle f|_{X}:=f\circ i}$.

In the situation one also calls a continuation of . An example of this is the continuous continuation . ${\displaystyle g=f|_{X}}$ ${\displaystyle f}$ ${\displaystyle g}$

example

${\displaystyle \mathbb {R} }$be the set of real numbers and with the square function . is not injective , but the restriction to the interval of non-negative real numbers is. If you also still the target amount to the image set (also restricts), we obtain the bijective quadratic function with , so a reversal function has, namely the square root function . ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle f(x)=x^{2}}$ ${\displaystyle f}$ ${\displaystyle f|_{S}}$ ${\displaystyle S:=[0,\infty )}$ ${\displaystyle S}$ ${\displaystyle g\colon S\to S}$ ${\displaystyle g(x)=x^{2}}$

Compatibility rules

The union of the (graphs of the) constraints of a function on a set and a set is equal to the constraint on the union of these two sets. The same applies to the cut: ${\displaystyle f}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$

${\displaystyle f|_{X_{1}}\cup \;f|_{X_{2}}=f_{X_{1}\cup X_{2}}}$

${\displaystyle f|_{X_{1}}\cap \;f|_{X_{2}}=f_{X_{1}\cap X_{2}}}$

The same applies to other set operations, as well as to infinite union and intersection. From this it follows: If the two sets and are disjoint , then so are the (graphs of) restricted functions and . ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle f|_{X_{1}}}$ ${\displaystyle f|_{X_{2}}}$

Restriction of a relation

Double-digit relations

Let be a two-digit relation from the pre-domain into the post-domain and be sets, then is called ${\displaystyle R\subseteq A\times B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle X,Y}$

${\displaystyle R\upharpoonleft X\equiv R|X:=R\cap (X\times Wb(R))=\{(a,b)\in R\mid a\in X\}}$

the pre-restriction of in and ${\displaystyle R}$ ${\displaystyle X}$

${\displaystyle R\upharpoonright Y:=R\cap (Db(R)\times Y)=\{(a,b)\in R\mid b\in Y\}}$

the subsequent restriction of in . In practice, is usually and apply, although not a requirement must be. ${\displaystyle R}$ ${\displaystyle Y}$ ${\displaystyle X\subseteq A}$ ${\displaystyle Y\subseteq B}$

If one takes the alternative, detailed definition of relations as a basis, then the pre- restriction of to a set turns out to be ${\displaystyle R=(G_{R},A,B)}$ ${\displaystyle G_{R}\equiv }$  ${\displaystyle \operatorname {Graph} (R)}$  ${\displaystyle \subseteq A\times B}$ ${\displaystyle R}$ ${\displaystyle X}$

${\displaystyle R\upharpoonleft X\equiv R|X:=(G_{R}\cap (X\times Wb(R)),A\cap X,B)}$

and the post-restriction to a lot as ${\displaystyle Y}$

${\displaystyle R\upharpoonright Y:=(G_{R}\cap (Db(R)\times Y),A,B\cap Y)}$.

As long as the definition or value ranges are not restricted ( or ), the pre- and post- restrictions are essentially the same as the original relation (especially in the case of equality ). ${\displaystyle X\supseteq Db(R)}$ ${\displaystyle Y\supseteq Wb(R)}$ ${\displaystyle X=Db(R),Y=Wb(R)}$

Homogeneous two-digit relations

In the case of homogeneous two-digit relations on the set (i.e. ) one speaks of a total restriction (or simply restriction ) if this relation is simultaneously pre-and post- restricted in the same set: ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle R\subseteq A\times A}$

${\displaystyle R\uparrow X\equiv R||X:=R\upharpoonleft X\upharpoonright X=R\cap (X\times X)=\{(a,b)\in R\mid a\in X\land b\in X\}}$

The order in which the pre-constraints and post-constraints are applied does not matter.
In particular, the following applies: If there is a homogeneous two-digit relation on the set and a subset of then the relation to the restriction from on if for all and off applies: ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A,}$ ${\displaystyle S}$ ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle X,}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle X}$

${\displaystyle a\;S\;b\Leftrightarrow a\;R\;b}$.

n -digit relations

In principle, the above definition can be extended to any number of -digit relations. For an n-ary homogeneous relation on a set (i.e. ) the (total) restriction is given by ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle R\subseteq A^{n}}$

${\displaystyle R\uparrow X:=R\cap X^{n}}$

In particular analogously to the above: Is a homogeneous -ary relations on an amount (i.e.,.. , And) a subset of , then the ary relation to the restriction of on if for all -membered sequences from the following applies: ${\displaystyle R}$ ${\displaystyle n}$ ${\displaystyle A}$ ${\displaystyle R\subseteq A^{n}}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle S}$ ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle X,}$ ${\displaystyle n}$ ${\displaystyle a_{1},\dotsc ,a_{n}}$ ${\displaystyle X}$

${\displaystyle (a_{1},\dotsc ,a_{n})\in S\Leftrightarrow (a_{1},\dotsc ,a_{n})\in R}$

example

The smaller relation on the set of integers is the restriction of the smaller relation on the set of rational numbers .

Restriction of a representation

A linear representation of a group on vector space is a homomorphism of in the general linear group . A restriction can be understood to mean two different constructions. ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle \rho }$ ${\displaystyle G}$ ${\displaystyle \operatorname {GL} (V)}$

• If is an invariant subspace , then one gets a restricted representation . ${\displaystyle U\subset V}$ ${\displaystyle G\to GL(U)}$
• If a sub-group , then is a representation of which with is referred to (for restriction). If there is no likelihood of confusion, to write only or also briefly It also uses the spelling and for the restriction of a representation (on) of at ${\displaystyle H\subset G}$ ${\displaystyle \rho |_{H}}$ ${\displaystyle H}$ ${\displaystyle \operatorname {Res} _{H}^{G}(\rho )}$ ${\displaystyle \operatorname {Res} (\rho )}$ ${\displaystyle \operatorname {Res} \rho .}$ ${\displaystyle \operatorname {Res} _{H}(V)}$ ${\displaystyle \operatorname {Res} (V)}$ ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle H.}$

literature

• Dieter Klaua : set theory . De Gruyter textbook. de Gruyter, Berlin, New York 1979, ISBN 3-11-007726-4 . The author uses the term correspondence in the set-theoretical sense synonymously with relation, but then uses the symbol instead of . In the article here, however, and (graph of ) is used throughout . ${\displaystyle F}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle G_{R}}$ ${\displaystyle R}$
• Willard van Orman Quine : Set Theory And Its Logic . Belknap Press of Harvard University Press, Cambridge, USA 1963, ISBN 0-674-80207-1 . P. 359 (HC) / 380 (PB).
  Willard van Orman Quine : Set theory and its logic (=  logic and foundations of mathematics (German translation) . Volume 10 ). Vieweg + Teubner Verlag, 1973, ISBN 3-528-08294-1 , pp. 264 . The author uses Greek lowercase letters to denote quantities in general (as here and ) and relations in particular. The page numbers refer to the German translation. ${\displaystyle X}$ ${\displaystyle Y}$

References and comments

1. ↑ ^{a b} Occasionally a different notation is used in set theory:

   ${\displaystyle R{\upharpoonright }_{X\times Y}=R\cap (X\times Y)}$

   ${\displaystyle R{\upharpoonright }_{X}\equiv R|_{X}=\{(x,y)\in R\mid x\in X\}}$

   and also for images (functions)

   ${\displaystyle f{\upharpoonright }_{X\times Y}=f\cap (X\times Y)}$

   ${\displaystyle f{\upharpoonright }_{X}\equiv f|_{X}=\{(x,y)\in f\mid x\in X\}}$

   For examples, see Proofwiki: Restriction , Proofwiki: Restriction / Mapping and Martin Ziegler: Lecture on Set Theory , University of Freiburg, 1992–2014, page 7. Note that this notation with the harpoon symbol is used in different ways and partly contrary to that of W . v. O. Quine and D. Klaua is!
2. ↑ D. Klaua: Set theory, p. 66, definition 8 (a), part 1 , part 2 , part 3 .
3. ↑ W. v. O. Quine: Set theory and its logic, page 47, 9.16 f.
4. ↑ There are

   ${\displaystyle Db(R)=\{a\in A\mid \exists b\in B\colon (a,b)\in R\},\ \;\ Wb(R)=\{b\in B\mid \exists a\in A\colon (a,b)\in R\}}$

   the definition and value range of the relation ; is the existential quantifier , read: There is (at least) one ... ${\displaystyle R}$ ${\displaystyle \exists }$
5. ↑ D. Klaua, set theory, p. 66, definition 8 (a), part 4.