Restriction (mathematics) explained

f is a new function, denoted f\vert _{A} or f {\restriction _{A},} obtained by choosing a smaller domain A for the original function f. The function f is then said to extend f\vert _{A.}

Formal definition

Let

f : E \to F be a function from a set E to a set F. If a set A is a subset of E, then the restriction of f to A is the function^[1] $$\_A\; :\; A\; \backslash to\; F$$given by {f|} _{A(x)} = f(x) for x \in A. Informally, the restriction of f to A is the same function as f, but is only defined on A .

If the function

f is thought of as a relation (x,f(x)) on the Cartesian product E x F, then the restriction of f to A can be represented by its graph,G({f|} _{A)} = \{ (x,f(x))\in G(f) : x\in A \} = G(f)\cap (A x F), where the pairs (x,f(x)) represent ordered pairs in the graph G.

Extensions

A function

F is said to be an of another function f if whenever x is in the domain of f then x is also in the domain of F and f(x) = F(x). That is, if \operatorname{domain} f \subseteq \operatorname{domain} F and F \ vert _{\operatorname{domain} f} = f.

A (respectively, , etc.) of a function

f is an extension of f that is also a linear map (respectively, a continuous map, etc.).

Examples

1. The restriction of the non-injective function

f: R \to R , x \mapsto x ^{2} to the domain R _{+} = [ 0 , inf ty) is the injectionf: R _{+} \to R , x \mapsto x ^{2.}

1. The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one:

{\Gamma|} _{Z ^{+}} (n) = (n -1 )!

Properties of restrictions

• Restricting a function

f:X → Y to its entire domain X gives back the original function, that is, f| _{X} = f.

• Restricting a function twice is the same as restricting it once, that is, if

A \subseteq B \subseteq \operatorname{dom} f, then \left(f| _{B\right)|} _{A} = f| _{A.}

• The restriction of the identity function on a set

X to a subset A of X is just the inclusion map from A into X. ^[2]

• The restriction of a continuous function is continuous.^{[3] [4]}

Applications

Inverse functions

See main article: Inverse function. For a function to have an inverse, it must be one-to-one. If a function

f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function$$f(x)\; =\; x^2$$defined on the whole of \R is not one-to-one since x ^{2} = ( - x) ^{2} for any x \in \R. However, the function becomes one-to-one if we restrict to the domain \R _{\geq} = [ 0 , inf ty), in which case$$f^(y)\; =\; \backslash sqrt\; .$$

(If we instead restrict to the domain

( -inf ty, 0 ], then the inverse is the negative of the square root of y. ) Alternatively, there is no need to restrict the domain if we allow the inverse to be a multivalued function.

Selection operators

See main article: Selection (relational algebra).

In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as

\sigma _{a} (R) or \sigma _{a} (R) where:a and b are attribute names,\theta is a binary operation in the set \{<, \leq, = , ≠ , \geq, >\}, v is a value constant,R is a relation.

The selection

\sigma _{a} (R) selects all those tuples in R for which \theta holds between the a and the b attribute.

The selection

\sigma _{a} (R) selects all those tuples in R for which \theta holds between the a attribute and the value v.

Thus, the selection operator restricts to a subset of the entire database.

The pasting lemma

See main article: Pasting lemma.

The pasting lemma is a result in topology that relates the continuity of a function with the continuity of its restrictions to subsets.

Let

X,Y be two closed subsets (or two open subsets) of a topological space A such that A = X \cup Y, and let B also be a topological space. If f: A \to B is continuous when restricted to both X and Y, then f is continuous.

This result allows one to take two continuous functions defined on closed (or open) subsets of a topological space and create a new one.

Sheaves

See main article: Sheaf theory.

Sheaves provide a way of generalizing restrictions to objects besides functions.

In sheaf theory, one assigns an object

F(U) in a category to each open set U of a topological space, and requires that the objects satisfy certain conditions. The most important condition is that there are restriction morphisms between every pair of objects associated to nested open sets; that is, if V\subseteq U, then there is a morphism \operatorname{res} _{V,U} : F(U) \to F(V) satisfying the following properties, which are designed to mimic the restriction of a function:

• For every open set

U of X, the restriction morphism \operatorname{res} _{U,U} : F(U) \to F(U) is the identity morphism on F(U).

• If we have three open sets

W \subseteq V \subseteq U, then the composite \operatorname{res} _{W,V} \circ \operatorname{res} _{V,U} = \operatorname{res} _{W,U} .

• (Locality) If

\left(U _{i\right)} is an open covering of an open set U, and if s, t \in F(U) are such that s \ vert _{U _{i}} = t \ vert _{U _{i}} for each set U _{i} of the covering, then s = t ; and

• (Gluing) If

\left(U _{i\right)} is an open covering of an open set U, and if for each i a section x _{i} \in F\left(U _{i\right)} is given such that for each pair U _{i,} U _{j} of the covering sets the restrictions of s _{i} and s _{j} agree on the overlaps: s _{i\vert} ^{U _{i} \cap U _{j}} = s _{j\vert} ^{U _{i} \cap U _{j}} , then there is a section s \in F(U) such that s \ vert _{U _{i}} = s _{i} for each i.

The collection of all such objects is called a sheaf. If only the first two properties are satisfied, it is a pre-sheaf.

Left- and right-restriction

More generally, the restriction (or domain restriction or left-restriction)

A \triangleleft R of a binary relation R between E and F may be defined as a relation having domain A, codomain F and graph G(A \triangleleft R) = \{(x, y) \in F(R) : x \in A\}. Similarly, one can define a right-restriction or range restriction R \triangleright B. Indeed, one could define a restriction to n -ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product E x F for binary relations.These cases do not fit into the scheme of sheaves.

Anti-restriction

The domain anti-restriction (or domain subtraction) of a function or binary relation

R (with domain E and codomain F ) by a set A may be defined as (E \setminus A) \triangleleft R ; it removes all elements of A from the domain E. It is sometimes denoted A  ⩤ R. ^[5] Similarly, the range anti-restriction (or range subtraction) of a function or binary relation R by a set B is defined as R \triangleright (F \setminus B) ; it removes all elements of B from the codomain F. It is sometimes denoted R  ⩥ B.

Notes and References

1. Book: Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. 1974. San Francisco. [36]. 2nd. 0-7167-0457-9.
2. Book: Halmos, Paul. Paul Halmos. Naive Set Theory. Princeton, NJ. D. Van Nostrand. 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. (Paperback edition).
3. Book: Munkres, James R.. Topology. 2nd. Upper Saddle River. Prentice Hall. 2000. 0-13-181629-2.
4. Book: Adams, Colin Conrad. Robert David. Franzosa. Introduction to Topology: Pure and Applied. Pearson Prentice Hall. 2008. 978-0-13-184869-6.
5. Dunne, S. and Stoddart, Bill Unifying Theories of Programming: First International Symposium, UTP 2006, Walworth Castle, County Durham, UK, February 5–7, 2006, Revised Selected ... Computer Science and General Issues). Springer (2006)