Restriction (mathematics)

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.