Find an infinite set $S$ and a function $g : S \to S$ that is
surjective but not injective.
This is all that is given in the problem. Should I fix $S$ to be a certain set, like the integers, or natural numbers, and work from there? Or should I just create a function, like $f(x) = x^2$? Any guidance would be appreciated.
What about this: Let $S$ be $\mathbb{R}$, and let $g : S \to S$ be defined by $f(x) = x^2$. This is surjective but not injective.
EDIT 11/27, 3:53pm CT: I totally confused myself about what it means to be surjective. I'm not sure what I was thinking. Clearly, $f(x) = x^2$ is not surjective since each y-coordinate is not mapped to.
Best Answer
Your proposed function won't work because it is not surjective. For example, $-1$ has no preimage but $-1 \in \mathbb{R}$.
Possible direction:
To make it work, for example, you can let $S$ be the set of nonnegative integer.
Let $g(0)=0=g(1)$, $g(2)=1=g(3)$, and et cetera.
Try to come out with a rule to describe what I did.