Functions:Restriction

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

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 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 $\sigma _{a\theta b}(R)$ 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 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 ∘ 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|_Ui = t|_Ui 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|_Ui∩Uj = s_j|_Ui∩Uj, then there is a section s ∈ F(U) such that s|_Ui = 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) ∈ 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.

Licensing

Content obtained and/or adapted from:

• Restriction (mathematics), Wikipedia under a CC BY-SA license