Restriction (mathematics)

Function

x \mapsto f (x)

By domain and codomain

$X$	—›	$B$			$B n$	—›	$B$
$X$	—›	$Z$	—›	$X$
$X$	—›	$R$	—›	$X$	$R n$	—›	$X$
$X$	—›	$C$	—›	$X$	$C n$	—›	$X$

Classes/properties

Constant · Identity · Linear · Polynomial · Rational · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective

Constructions

Restriction · Composition · λ · Inverse

Generalizations

Partial · Multivalued · Implicit

The function x² with domain R does not have an inverse. If we restrict x² to the non-negative real numbers, then it does have an inverse, known as the square root of x.

In mathematics, the restriction of a function f is a new function $f\vert _{A}$ obtained by choosing a smaller domain A for the original function $f$ . The notation $f{\upharpoonright _{A}}$ is also used.

Formal definition

Let $f:E\to F$ be a function from a set $E$ to a set $F$ , so that the domain of $f$ is in $E$ ( ${\mathrm {dom}}\,f\subseteq E$ ). If a set $A$ is a subset of $E$ , then the restriction of $f$ to $A$ is the function^[1]

{f|}_{A}\colon A\to F

Informally, the restriction of $f$ to $A$ is the same function as $f$ , but is only defined on $A\cap {\mathrm {dom}}\,f$ .

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

Examples

The restriction of the non-injective function $f:{\mathbb R}\to {\mathbb R};x\mapsto x^{2}$ to ${\mathbb R}_{+}=[0,\infty )$ is the injection $f:{\mathbb R}_{+}\to {\mathbb R};x\mapsto x^{2}$ .
The factorial function is the restriction of the gamma function to the integers.

Properties of restrictions

Restricting a function $f:X\rightarrow Y$ to its entire domain $X$ gives back the original function; i.e., $f|_{{X}}=f$ .
Restricting a function twice is the same as restricting it once; i.e. if $A\subseteq B\subseteq {\mathrm {dom}}f$ , then $(f|_{B})|_{A}=f|_{A}$ .
The restriction of the identity function on a space X to a subset A of X is just the inclusion map of A into X.^[2]
The restriction of a continuous function is continuous.^[3]^[4]

Applications

Inverse functions

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}

is not one-to-one, since $x 2 = (- x) 2$ . However, the function becomes one-to-one if we restrict to the domain $x \geq 0$ , in which case

f^{-1}(y)={\sqrt {y}}.

(If we instead restrict to the domain $x \leq 0$ , then the inverse is the negative of the square root of $y$ .) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:

Selection operators

In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as $\sigma _{a\theta b}(R)$ or $\sigma _{a\theta v}(R)$ where:

$a$ and $b$ are attribute names
$\theta$ is a binary operation in the set $\{\;<,\leq ,=,\neq ,\geq ,\;>\}$
$v$ is a value constant
$R$ is a relation

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

The selection $\sigma _{a\theta v}(R)$ 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

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 both closed (or both 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

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; i.e., if $V\subseteq U$ , then there is a morphism res_V,U : F(U) → 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 res_U,U : F(U) → F(U) is the identity morphism on F(U).
If we have three open sets W ⊆ V ⊆ U, then the composite res_W,V o res_V,U = res_W,U.
(Locality) If (U_i) is an open covering of an open set U, and if s,t ∈ F(U) are such that s|_{U_i} = t|_{U_i} for each set U_i of the covering, then s = t; and
(Gluing) If (U_i) is an open covering of an open set U, and if for each i a section s_i ∈ F(U_i) 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|_{U_i∩U_j} = s_j|_{U_i∩U_j}, then there is a section s ∈ F(U) such that s|_{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) $A ◁ R$ of a binary relation $R$ between $E$ and $F$ may be defined as a relation having domain $A$ , codomain $F$ and graph $G(A ◁ R) = {(x, y) \in G(R) | x \in A}$ . Similarly, one can define a right-restriction or range restriction $R ▷ B$ . Indeed, one could define a restriction to $n$ -ary relations, as well as to subsets understood as relations, such as ones of $E \times F$ 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 $(E \ A) ◁ R$ ; 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 $R ▷ (F \ B)$ ; it removes all elements of $B$ from the codomain $F$ . It is sometimes denoted $R$ ⩥ $B$ .

References

↑ Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. p. 5.
↑ Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition).
↑ Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
↑ Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.
↑ 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)

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