# Partial Function

A Partial Function is a Function that is not defined for all Elements in its Input Set.

**Example(s):****See:**Lazy Learning, Surjective Function.

## References

### 2014

- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/partial_function#Total_function Retrieved:2014-4-21.
- In mathematics, a
**partial function**from*X*to Y*(written as*f: X ↛ Y*) is a function*f: X' → Y, where*X'*is a subset 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 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:**f*(*x*) =*y*∈*Y*(it is defined as a single element in*Y*) or*f*(*x*) is undefined.

- For example we can consider the square root function restricted to the integers :[math]g\colon \mathbb{Z} \to \mathbb{Z}[/math] :[math]g(n) = \sqrt{n}.[/math]
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.*

- In mathematics, a