 Restriction (mathematics)
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Restriction Mathematics 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, denoted $f\vert _{A}$ or $f{\upharpoonright _{A}}$ , obtained by choosing a smaller domain A for the original function $f$ .

## 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

${f|}_{A}\colon A\to F$ 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 $A\cap \operatorname {dom} f$ .

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 its 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 ordered pairs in the graph G.

## Examples

1. The restriction of the non-injective function$f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto x^{2}$ to the domain $\mathbb {R} _{+}=[0,\infty )$ is the injection$f:\mathbb {R} _{+}\to \mathbb {R} ,\ x\mapsto x^{2}$ .
2. The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one: ${\Gamma |}_{\mathbb {Z} ^{+}}\!(n)=(n-1)!$ ## 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 $A\subseteq B\subseteq \operatorname {dom} f$ , then $(f|_{B})|_{A}=f|_{A}$ .
• 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.
• The restriction of a continuous function is continuous.

## 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

$f(x)=x^{2}$ defined on the whole of $\mathbb {R}$ is not one-to-one since x2 = (-x)2 for any x in $\mathbb {R}$ . However, the function becomes one-to-one if we restrict to the domain $\mathbb {R} _{\geq 0}=[0,\infty )$ , in which case

$f^{-1}(y)={\sqrt {y}}.$ (If we instead restrict to the domain $(-\infty ,0]$ , then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we don't mind 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 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

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 resV,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 resU,U : F(U) -> F(U) is the identity morphism on F(U).
• If we have three open sets W ? V ? U, then the composite
• (Locality) If (Ui) is an open covering of an open set U, and if s,t ? F(U) are such that s|Ui = t|Ui for each set Ui of the covering, then s = t; and
• (Gluing) If (Ui) is an open covering of an open set U, and if for each i a section si ? F(Ui) is given such that for each pair Ui,Uj of the covering sets the restrictions of si and sj agree on the overlaps: si|Ui?Uj = sj|Ui?Uj, then there is a section s ? F(U) such that s|Ui = si 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) ? G(R) | x ? 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 × F for binary relations. These cases do not fit into the scheme of sheaves.[clarification needed]

## 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. 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.