Ads
related to: define domain restriction
Search results
Results From The WOW.Com Content Network
More generally, the restriction (or domain restriction or left-restriction) of a binary relation between and may be defined as a relation having domain , codomain and graph ( ) = {(,) ():}. Similarly, one can define a right-restriction or range restriction R B . {\displaystyle R\triangleright B.}
Restrictions can be used to define partial inverse functions: if there is a subset S of the domain of a function such that | is injective, then the canonical surjection of | onto its image | = is a bijection, and thus has an inverse function from () to S.
Also, the "member of" relation needs to be restricted to have domain and codomain () to obtain a binary relation that is a set. Bertrand Russell has shown that assuming ∈ {\displaystyle \in } to be defined over all sets leads to a contradiction in naive set theory , see Russell's paradox .
The set of points in the red oval X is the domain of f. Graph of the real-valued square root function, f(x) = √ x, whose domain consists of all nonnegative real numbers. In mathematics, the domain of a function is the set of inputs accepted by the function.
Interpretation for surjective functions in the Cartesian plane, defined by the mapping f : X → Y, where y = f(x), X = domain of function, Y = range of function. Every element in the range is mapped onto from an element in the domain, by the rule f. There may be a number of domain elements which map to the same range element.
The domain of definition of a partial function is the subset S of X on which the partial function is defined; in this case, the partial function may also be viewed as a function from S to Y. In the example of the square root operation, the set S consists of the nonnegative real numbers [ 0 , + ∞ ) . {\displaystyle [0,+\infty ).}
The set of these functions, together with an appropriate ordering, is again a "domain" in the sense of the theory. But the restriction to a subset of all available functions has another great benefit: it is possible to obtain domains that contain their own function spaces, i.e. one gets functions that can be applied to themselves.
A function defined on a rectangle (top figure, in red), and its trace (bottom figure, in red). In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space.