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:
“In the domain of art there is no light without heat.”
—Victor Hugo (1802–1885)
“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)
“And meanwhile we have gone on living,
Living and partly living,
Picking together the pieces,
Gathering faggots at nightfall,
Building a partial shelter,
For sleeping and eating and drinking and laughter.”
—T.S. (Thomas Stearns)
“The more books we read, the clearer it becomes that the true function of a writer is to produce a masterpiece and that no other task is of any consequence.”
—Cyril Connolly (1903–1974)