Partial function

In mathematics, a partial function from X to Y (sometimes written as f : X ↛ Y, f: X ⇸ Y, or f: X ↪ Y) is a function f: X′ → Y, for some subset X′ of X. It generalizes the concept of a function f : X → Y by not forcing f to map every element of X to an element of Y (only some subset X′ of X). If X′ = X, then f is called a total function for emphasizing that its domain is not a proper subset of X. Partial functions are often used when the exact domain, X, is not known (for example, in computability theory, general recursive functions are partial functions from the integers to the integers, and there cannot be any algorithm for deciding whether such a function is total). In real and complex analysis, a partial function is generally called simply a function. In mathematics, a partial function from X to Y (sometimes written as f : X ↛ Y, f: X ⇸ Y, or f: X ↪ Y) is a function f: X′ → Y, for some subset X′ of X. It generalizes the concept of a function f : X → Y by not forcing f to map every element of X to an element of Y (only some subset X′ of X). If X′ = X, then f is called a total function for emphasizing that its domain is not a proper subset of X. Partial functions are often used when the exact domain, X, is not known (for example, in computability theory, general recursive functions are partial functions from the integers to the integers, and there cannot be any algorithm for deciding whether such a function is total). In real and complex analysis, a partial function is generally called simply a function. Specifically, we will say that for any x ∈ X, either: For example, we can consider the square root function restricted to the integers Thus g(n) is only defined for n that are perfect squares (i.e., 0, 1, 4, 9, 16, ...). So, g(25) = 5, but g(26) is undefined. 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. Similarly, the term range can refer to either the codomain or the image of a function. 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 injective or surjective respectively. A partial function may be both injective and surjective (and thus bijective). Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective.