Restriction (mathematics)

In mathematics, the restriction of a function ${\displaystyle f}$ is a new function, denoted ${\displaystyle f\vert _{A}}$ or ${\displaystyle f{\upharpoonright _{A}}}$, obtained by choosing a smaller domain A for the original function ${\displaystyle f}$.

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.

Formal definition

Let ${\displaystyle f:E\to F}$ be a function from a set E to a set F. If a set A is a subset of E, then the restriction of ${\displaystyle f}$ to ${\displaystyle A}$ is the function[1]

${\displaystyle {f|}_{A}\colon A\to F}$

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 ${\displaystyle A\cap \operatorname {dom} f}$.

If the function f is thought of as a relation ${\displaystyle (x,f(x))}$ on the Cartesian product ${\displaystyle E\times F}$, then the restriction of f to A can be represented by its graph${\displaystyle G({f|}_{A})=\{(x,f(x))\in G(f)\mid x\in A\}=G(f)\cap (A\times F)}$, where the pairs ${\displaystyle (x,f(x))}$ represent ordered pairs in the graph G.

Examples

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

Properties of restrictions

• Restricting a function ${\displaystyle f:X\rightarrow Y}$ to its entire domain ${\displaystyle X}$ gives back the original function, i.e., ${\displaystyle f|_{X}=f}$.
• Restricting a function twice is the same as restricting it once, i.e. if ${\displaystyle A\subseteq B\subseteq \operatorname {dom} f}$, then ${\displaystyle (f|_{B})|_{A}=f|_{A}}$.
• 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

Main article: Inverse function

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

${\displaystyle f(x)=x^{2}}$

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

${\displaystyle f^{-1}(y)={\sqrt {y}}.}$

(If we instead restrict to the domain ${\displaystyle (-\infty ,0]}$, 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

Main article: Selection (relational algebra)

In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as ${\displaystyle \sigma _{a\theta b}(R)}$ or ${\displaystyle \sigma _{a\theta v}(R)}$ where:

• ${\displaystyle a}$ and ${\displaystyle b}$ are attribute names,
• ${\displaystyle \theta }$ is a binary operation in the set ${\displaystyle \{<,\leq ,=,\neq ,\geq ,>\}}$,
• ${\displaystyle v}$ is a value constant,
• ${\displaystyle R}$ is a relation.

The selection ${\displaystyle \sigma _{a\theta b}(R)}$ selects all those tuples in ${\displaystyle R}$ for which ${\displaystyle \theta }$ holds between the ${\displaystyle a}$ and the ${\displaystyle b}$ attribute.

The selection ${\displaystyle \sigma _{a\theta v}(R)}$ selects all those tuples in ${\displaystyle R}$ for which ${\displaystyle \theta }$ holds between the ${\displaystyle a}$ attribute and the value ${\displaystyle v}$.

Thus, the selection operator restricts to a subset of the entire database.

The pasting lemma

Main article: 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 ${\displaystyle X,Y}$ be two closed subsets (or two open subsets) of a topological space ${\displaystyle A}$ such that ${\displaystyle A=X\cup Y}$, and let ${\displaystyle B}$ also be a topological space. If ${\displaystyle f:A\to B}$ is continuous when restricted to both ${\displaystyle X}$ and ${\displaystyle Y}$, then ${\displaystyle f}$ 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 ${\displaystyle F(U)}$ in a category to each open set ${\displaystyle U}$ 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 ${\displaystyle V\subseteq U}$, 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 o 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.

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.

See also

• Constraint
• Deformation retract
• Function (mathematics) § Restriction and extension
• Binary relation § Restriction
• Relational algebra § Selection (σ)

References

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)