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:
“Every sign is subject to the criteria of ideological evaluation.... The domain of ideology coincides with the domain of signs. They equate with one another. Wherever a sign is present, ideology is present, too. Everything ideological possesses semiotic value.”
—V.N. (Valintin Nikolaevic)
“While you are divided from us by geographical lines, which are imaginary, and by a language which is not the same, you have not come to an alien people or land. In the realm of the heart, in the domain of the mind, there are no geographical lines dividing the nations.”
—Anna Howard Shaw (1847–1919)
“You must not be partial in judging: hear out the small and the great alike; you shall not be intimidated by anyone, for the judgment is God’s.”
—Bible: Hebrew, Deuteronomy 1:17.
““... The state’s one function is to give.
The bud must bloom till blowsy blown
Its petals loosen and are strown;
And that’s a fate it can’t evade
Unless ‘twould rather wilt than fade.””
—Robert Frost (1874–1963)