Restriction (mathematics)

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 $f.$ 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] ${f|}_{A}:A\to F$ 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 $A$ .

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):x\in 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 extension 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 $F{\big \vert }_{\operatorname {domain} f}=f.$

A linear extension (respectively, continuous extension, 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 $f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto x^{2}$ to the domain $\mathbb {R} _{+}=[0,\infty )$ is the injection $f:\mathbb {R} _{+}\to \mathbb {R} ,\ x\mapsto x^{2}.$
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, 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

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 $x^{2}=(-x)^{2}$ 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; 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 $s{\big \vert }_{U_{i}}=t{\big \vert }_{U_{i}}$ 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: $s_{i}{\big \vert }_{U_{i}\cap U_{j}}=s_{j}{\big \vert }_{U_{i}\cap U_{j}},$ then there is a section $s\in F(U)$ such that $s{\big \vert }_{U_{i}}=s_{i}$ 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 $E\times F$ 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 $(E\setminus A)\triangleleft R$ ; 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 $R$ ⩥ $B.$

References

^ Stoll, Robert (1974). Sets, Logic and Axiomatic Theories (2nd ed.). San Francisco: W. H. Freeman and Company. pp. [36]. ISBN 0-7167-0457-9.
^ Halmos, Paul (1960). Naive Set Theory. Princeton, NJ: D. Van Nostrand. 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. (2000). Topology (2nd ed.). Upper Saddle River: Prentice Hall. ISBN 0-13-181629-2.
^ Adams, Colin Conrad; Franzosa, Robert David (2008). Introduction to Topology: Pure and Applied. Pearson Prentice Hall. ISBN 978-0-13-184869-6.
^ 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)

[Stoll-1] Stoll, Robert (1974). Sets, Logic and Axiomatic Theories (2nd ed.). San Francisco: W. H. Freeman and Company. pp. [36]. ISBN 0-7167-0457-9.

[2] Halmos, Paul (1960). Naive Set Theory. Princeton, NJ: D. Van Nostrand. 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).

[3] Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River: Prentice Hall. ISBN 0-13-181629-2.

[4] Adams, Colin Conrad; Franzosa, Robert David (2008). Introduction to Topology: Pure and Applied. Pearson Prentice Hall. ISBN 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)

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