# Function that Returns a Set

I was having a talk with one of my computer science teachers and he claimed that there are no functions that can return a variable number of parameters in mathematics (outside of extremely abstract fields). I haven't had much experience with these topics, so instead I took this as a challenge and decided to try the following:

f:Z->{Z} f(x)={yϵZ|0<y<x}

Where 'f' should return the set of all integers in between the 0 and 'x' that aren't negative.

I understand that this isn't STRICTLY defining a function with multiple return values (such as a potential function that solves the quadratic equation), but is it still a series of valid mathematical statements that would provide a workaround counterexample?

(also, any information about functions that return variable-size tuples would be helpful)

EDIT: I might've overstated my professor's position on this, so please don't take that so far into consideration...

• For the last point: consider a function $f\colon X\to Y$, where $Y= \bigcup_{d=1}^\infty \mathbb{N}^d$. Nov 2, 2017 at 17:57
• Your professor overstated the case. He probably should have said something more like "You don't often see such a function in algebra or calculus." Nov 2, 2017 at 17:58
• In ZFC everything is a set, so all functions return sets. Nov 2, 2017 at 18:08
• Ooh, not really. That has returned one set which is a single parameter. I'd say further than all functions return one parameter. Which intern may be a set of variable elements so... I don't think the professor is being as deep as he thinks he is. I don't want to be cynical or dismissive but I'd say the professors statement is not worth stating. And I think you did a good job showing the statement wasn't that deep or perplexing with thought. Yes, I'd call yours a counter-example to the statement having any significance, but not to it being literally true. Nov 2, 2017 at 18:30
• 99 times out of 100, when someone says "mathematics doesn't do X" or "mathematics can't do Y" or something similar, what is actually going on is either (1) "mathematics doesn't do X and Y in the very specific way that I imagined, and I never really put much thought as to how one might do it differently" or (2) "While I'm using the words, I don't mean what they usually mean, or only mean for my assertion to be applied to a narrow setting and I forgot to actually tell you what that setting is".
– user14972
Nov 2, 2017 at 19:30

$$f: X \to 2^Y.$$
Here, $$f$$ maps each member of $$X$$ into a subset of $$Y$$. ($$2^Y$$ denotes, as usual, the power set of $$Y$$.) So it is strictly a function that maps $$X$$ into $$2^Y$$. Of course, it is not a function from from $$X$$ into $$Y$$. (If $$f$$ were singleton-valued, however, we can make the obvious identification of $$f$$ with a function from $$X$$ into $$Y$$.)
In fact, I think you point out a good example of a set-valued function. Suppose $$f:\mathbb R\times\mathbb R \times \mathbb R \to 2^{\mathbb C},$$ where $$f$$ takes the real numbers $$(a,b,c)$$ and returns the roots of the quadratic $$ax^2 +bx + c = 0$$. $$f$$ is clearly well-defined as a set-valued function on $$\mathbb C$$, but you'd need to do some extra work if want $$f$$ to be a classical function.