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Concept
Restriction (mathematics)
Summary
In mathematics, the restriction of a function is a new function, denoted or obtained by choosing a smaller domain for the original function
The function is then said to extend
Let be a function from a set to a set If a set is a subset of then the restriction of to is the function
given by for Informally, the restriction of to is the same function as but is only defined on .
If the function is thought of as a relation on the Cartesian product then the restriction of to can be represented by its graph where the pairs represent ordered pairs in the graph
A function is said to be an of another function if whenever is in the domain of then is also in the domain of and
That is, if and
A (respectively, , etc.) of a function is an extension of that is also a linear map (respectively, a continuous map, etc.).
The restriction of the non-injective function to the domain is the injection
The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one:
Restricting a function to its entire domain gives back the original function, that is,
Restricting a function twice is the same as restricting it once, that is, if then
The restriction of the identity function on a set to a subset of is just the inclusion map from into
The restriction of a continuous function is continuous.
Inverse function
For a function to have an inverse, it must be one-to-one. If a function is not one-to-one, it may be possible to define a partial inverse of by restricting the domain. For example, the function
defined on the whole of is not one-to-one since for any However, the function becomes one-to-one if we restrict to the domain in which case
(If we instead 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.
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
or where:
and are attribute names,
is a binary operation in the set
is a value constant,
is a relation.
Official source
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