Restriction (mathematics)

In mathematics, the restriction of a function

obtained by choosing a smaller domain

for the original function

is thought of as a relation

can be represented by its graph, where the pairs

represent ordered pairs in the graph

For a function to have an inverse, it must be one-to-one.

is not one-to-one, it may be possible to define a partial inverse of

However, the function becomes one-to-one if we restrict to the domain

then the inverse is the negative of the square root of

) Alternatively, there is no need to restrict the domain if we allow the inverse to be a multivalued function.

In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as

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

The pasting lemma is a result in topology that relates the continuity of a function with the continuity of its restrictions to subsets.

be two closed subsets (or two open subsets) of a topological space

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 provide a way of generalizing restrictions to objects besides functions.

In sheaf theory, one assigns an object

in a category to each open set

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

satisfying the following properties, which are designed to mimic the restriction of a function: The collection of all such objects is called a sheaf.

If only the first two properties are satisfied, it is a pre-sheaf.

More generally, the restriction (or domain restriction or left-restriction)

may be defined as a relation having domain

Similarly, one can define a right-restriction or range restriction

-ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product

These cases do not fit into the scheme of sheaves.

[clarification needed] The domain anti-restriction (or domain subtraction) of a function or binary relation

[5] Similarly, the range anti-restriction (or range subtraction) of a function or binary relation