In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function 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 ƒ is called a total function and is equivalent to a function. Partial functions are often used when the exact domain, X', is not known (e.g. many functions in computability theory).
Specifically, we will say that for any x ∈ X, either:
- ƒ(x) = y ∈ Y (it is defined as a single element in Y) or
- ƒ(x) is undefined.
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.
Read more about Partial Function: Domain of A Partial Function, Total Function, Discussion and Examples
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