Restriction of a function

In mathematics, the restriction of a function $f$ is a new function, denoted $f\vert_A$ or $f ,$ 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

$_A : A \to F$
given by $_A(x) = f(x)$ 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 $E \times F,$ then the restriction of $f$ to $A$ can be represented by its graph,
:$G(_A) = \ = G(f)\cap (A\times F),$
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 $\operatorname f \subseteq \operatorname F$ and $F\big\vert_ = f.$

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

Examples

# The restriction of the non-injective function$f: \mathbb \to \mathbb, \ x \mapsto x^2$ to the domain \mathbb_ = ,\infty) is the injection$f:\mathbb_+ \to \mathbb, \ x \mapsto x^2.$
# The factorial function is the restriction of the gamma function">factorial">,\infty) is the injection$f:\mathbb_+ \to \mathbb, \ x \mapsto x^2.$
# The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one: $_\!(n) = (n-1)!$

Properties of restrictions

* Restricting a function $f:X\rightarrow Y$ 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 $A \subseteq B \subseteq \operatorname f,$ 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 $X.$
* 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 $\R$ 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_ = , \infty), in which case
$f^(y) = \sqrt .$

(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 allow the inverse to be 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
$\sigma_(R)$ or $\sigma_(R)$ where:
* $a$ and $b$ are attribute names,
* $\theta$ is a binary operation in the set $\,$
* $v$ is a value constant,
* $R$ is a relation.

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

The selection $\sigma_(R)$ 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; that is, if $V\subseteq U,$ then there is a morphism $\operatorname_ : F(U) \to F(V)$ satisfying the following properties, which are designed to mimic the restriction of a function:
* For every open set $U$ of $X,$ the restriction morphism $\operatorname_ : F(U) \to F(U)$ is the identity morphism on $F(U).$
* If we have three open sets $W \subseteq V \subseteq U,$ then the composite $\operatorname_ \circ \operatorname_ = \operatorname_.$
* (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 $s\big\vert_ = t\big\vert_$ 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: $s_i\big\vert_ = s_j\big\vert_,$ then there is a section $s \in F(U)$ such that $s\big\vert_ = s_i$ 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 $G(A \triangleleft R) = \.$ 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 $E \times F$ 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 $(E \setminus A) \triangleleft R$; it removes all elements of $A$ from the domain $E.$ It is sometimes denoted $A$ ⩤ $R.$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) 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 $R$ ⩥ $B.$

See also

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References

{{DEFAULTSORT:Restriction (Mathematics)

Sheaf theory