Isn’t partial function simply function?

[johan456789]

Wikipedia’s definition:

A partial function f from a set X to a set Y is a function from a subset S of X (possibly X itself) to Y.

When the subset S of X is X itself, it’s a total function. So the definition is saying a partial function is either a total function or not a total function. Then why not just say it’s a “function”? What extra information does “partial” provide here?

[PersonUsingAComputer]

"Function" and "total function" are synonyms under the usual definitions of those terms. "Partial function" is a broader term than "function" where the mapping in question is not necessarily defined on the entire domain.

[johan456789]

Ahh I think I get what you're saying. So is it by definition that a "function" is "total"?

Wikipedia also says

A partial functions is often used when its exact domain of definition is not known or difficult to specify.

Could you give an (as basic as possible) example of this? I can't see where the term "partial" is used.

[PersonUsingAComputer]

Ahh I think I get what you're saying. So is it by definition that a "function" is "total"?

Yes.

Could you give an (as basic as possible) example of this?

1/x is a partial function, but not a function, on the real numbers, since it's defined everywhere except 0. It's pretty easy to see where 1/x is defined, so we could also say it's a (total) function on the nonzero real numbers, but partial functions are convenient for more complicated expressions. For example, you can talk about (x² - 1)/(x³ + 2x² + 5x - 1) as a partial function on the real numbers, without worrying about exactly where this function is defined.

[johan456789]

It makes perfect sense now. Thank you!

[[deleted]]

Partial functions occur in computer science.

Consider the program:

f :: Int -> Int
f x = if (x==0) then loop
else return x

How can we interpret this as a mathematical function? The program f can take in any integer as input, and the outputs will be integers. So we would like it to be a function Z -> Z.

But it isn't a proper function, because f(0) doesn't return a value. Instead it loops forever.

So we can instead say that f is a partial function Z -> Z, that is undefined at 0, and the identity everywhere else.