Domain of A Partial Function
There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function. Most mathematicians, including recursion theorists, use the term "domain of f" for the set of all values x such that f(x) is defined ( X' above). But some, particularly category theorists, consider the domain of a partial function f:X→Y to be X, and refer to X' as the domain of definition.
Occasionally, a partial function with domain X and codomain Y is written as f: X ⇸ Y, using an arrow with vertical stroke.
A partial function is said to be injective or surjective when the total function given by the restriction of the partial function to its domain of definition is. A partial function may be both injective and surjective, but the term bijection generally only applies to total functions.
An injective partial function may be inverted to an injective partial function, and a partial function which is both injective and surjective has an injective function as inverse.
Read more about this topic: Partial Function
Famous quotes containing the words domain of, domain, partial and/or function:
“The vice named surrealism is the immoderate and impassioned use of the stupefacient image or rather of the uncontrolled provocation of the image for its own sake and for the element of unpredictable perturbation and of metamorphosis which it introduces into the domain of representation; for each image on each occasion forces you to revise the entire Universe.”
—Louis Aragon (1897–1982)
“You are the harvest and not the reaper
And of your domain another is the keeper.”
—John Ashbery (b. 1927)
“We were soon in the smooth water of the Quakish Lake,... and we had our first, but a partial view of Ktaadn, its summit veiled in clouds, like a dark isthmus in that quarter, connecting the heavens with the earth.”
—Henry David Thoreau (1817–1862)
“We are thus able to distinguish thinking as the function which is to a large extent linguistic.”
—Benjamin Lee Whorf (1897–1934)