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 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.

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}\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

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, 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.[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 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 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 $W \subseteq V \subseteq U$ , then the composite $res W, V \circ res V, U = res W, U$ .
(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 $s i \in F (U i)$ 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: $s i | U i \cap U j = s j | U i \cap U j$ , then there is a section $s \in F (U)$ 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 (1974). Sets, Logic and Axiomatic Theories (2nd ed.). San Francisco: W. H. Freeman and Company. pp. 5. 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)

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[Stoll-1] Stoll, Robert (1974). Sets, Logic and Axiomatic Theories (2nd ed.). San Francisco: W. H. Freeman and Company. pp. 5. 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)

Function
$x \mapsto f (x)$
Examples by domain and codomain
$X$ → $B,$ $B$ → $X,$ $B n$ → $B$ $X$ → $Z,$ $Z$ → $X$ $X$ → $R,$ $R$ → $X,$ $R n$ → $X$ $X$ → $C,$ $C$ → $X,$ $C n$ → $X$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Partial Multivalued Implicit