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 rightunambiguous, lefttotal relation, then this definition reproduces that of the prerestriction . With the help of the inclusion map, the restriction can be briefly written as
 .
In the situation one also calls a continuation of . An example of this is the continuous continuation .
example
be the set of real numbers and with the square function . is not injective , but the restriction to the interval of nonnegative 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 .
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:
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 .
Restriction of a relation
Doubledigit relations
Let be a twodigit relation from the predomain into the postdomain and be sets, then is called
the prerestriction of in and
the subsequent restriction of in . In practice, is usually and apply, although not a requirement must be.
If one takes the alternative, detailed definition of relations as a basis, then the pre restriction of to a set turns out to be
and the postrestriction to a lot as
 .
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 ).
Homogeneous twodigit relations
In the case of homogeneous twodigit relations on the set (i.e. ) one speaks of a total restriction (or simply restriction ) if this relation is simultaneously preand post restricted in the same set:
The order in which the preconstraints and postconstraints are applied does not matter.
In particular, the following applies: If there is a homogeneous twodigit relation on the set and a subset of then the relation to the restriction from on if for all and off applies:
 .
n digit relations
In principle, the above definition can be extended to any number of digit relations. For an nary homogeneous relation on a set (i.e. ) the (total) restriction is given by
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:
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.
 If is an invariant subspace , then one gets a restricted representation .
 If a subgroup , 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
literature
 Dieter Klaua : set theory . De Gruyter textbook. de Gruyter, Berlin, New York 1979, ISBN 3110077264 . The author uses the term correspondence in the settheoretical sense synonymously with relation, but then uses the symbol instead of . In the article here, however, and (graph of ) is used throughout .

Willard van Orman Quine : Set Theory And Its Logic . Belknap Press of Harvard University Press, Cambridge, USA 1963, ISBN 0674802071 . 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 3528082941 , 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.
References and comments

↑ ^{a } ^{b} Occasionally a different notation is used in set theory:
 ↑ D. Klaua: Set theory, p. 66, definition 8 (a), part 1 , part 2 , part 3 .
 ↑ W. v. O. Quine: Set theory and its logic, page 47, 9.16 f.

↑ There are
 ↑ D. Klaua, set theory, p. 66, definition 8 (a), part 4.