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...

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    $\begingroup$ For the last point: consider a function $f\colon X\to Y$, where $Y= \bigcup_{d=1}^\infty \mathbb{N}^d$. $\endgroup$
    – Clement C.
    Nov 2, 2017 at 17:57
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    $\begingroup$ 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." $\endgroup$ Nov 2, 2017 at 17:58
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    $\begingroup$ In ZFC everything is a set, so all functions return sets. $\endgroup$ Nov 2, 2017 at 18:08
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    $\begingroup$ 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. $\endgroup$
    – fleablood
    Nov 2, 2017 at 18:30
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    $\begingroup$ 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". $\endgroup$
    – user14972
    Nov 2, 2017 at 19:30

1 Answer 1


I think the object that you are trying to describe is called a set-valued function, and they are quite common in, for example, economics (though economists prefer to call them correspondences).

Formally, we have a function

$$ f: X \to 2^Y. $$

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.


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