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.
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