Restriction (mathematics)

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.

In mathematics, the restriction of a function f is a new function f|_A obtained by choosing a smaller domain A for the original function f. The notation $f {\restriction_A}$ is also used.

Formal definition

Let f : E → F be a function from a set E to a set F, so that the domain of f is in E ($\mathrm{dom} \, f \subseteq E$). If a set A is a subset of E, then the restriction of f to A is the function^[1]

${f|}_A \colon A \to F$.

Informally, the restriction of f to A is the same function as f, but is only defined on $A\cap \mathrm{dom} \, f$.

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 the 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 edges in the graph G.

Examples

1. The restriction of the non-injective function $f: \mathbb R\to\mathbb R; x\mapsto x^2$ to $\mathbb R_+=[0,\infty)$ 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 integers.

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 $A\subseteq B \subseteq \mathrm{dom} f$, then $(f|_B)|_A=f|_A$.
• The restriction of the identity function on a space X to a subset A of X is just the inclusion map of 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

$f(x) = x^2$

is not one-to-one, since x² = (−x)². However, the function becomes one-to-one if we restrict to the domain x ≥ 0, in which case

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

(If we instead restrict to the domain x ≤ 0, then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we are content with 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 $\{\;<, \le, =, \ne, \ge, \;>\}$
• $v$ 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 both closed (or both 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 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

• Deformation retract
• Function (mathematics) #Restrictions and extensions
• Binary relation #Restriction
• Relational algebra#Selection (σ)

References

1. Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. p. 5.
2. Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. 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. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
4. Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.
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)