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I’ve gotten a different answer than the one provided by the professor, and based on how the question is worded and the definition of onto, the answer ought to be no.

Textbook definition of onto: “A function f from A to B is called onto, or a surjection, if and only if for every element b ∈ B there is an element a ∈ A with f(a) = b. A function f is called surjective if it is onto.”

This is ℤ: {…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}

Both and () are in ℤ.

If is onto, then for every element in ℤ, there should be a corresponding in ℤ that produces it when fed to the function . However, the output of is only even integers—excludes odd integers. For that reason, cannot be onto because there are some elements in ℤ that are not outputs of .

() Comment
-1 -5
❌ -4 no corresponding input in ℤ
0 -3
❌ -2 no corresponding input in ℤ
1 -1
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Kevin Li
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    $2m-3$ is odd – J. W. Tanner Apr 19 '21 at 01:32
  • Here is a MathJax tutorial – J. W. Tanner Apr 19 '21 at 02:25
  • The professor said it was onto? Then professor is wrong as no even integers are mapped to it is not onto. But are you describing the question correctly? Is the question "Is $f: \mathbb Z \to \mathbb Z$ one-to-one?" (Answer: Yes.) Or is the question "Is $f:\mathbb Z \to \mathbb O = {z\in \mathbb Z| z\text{ is odd}}$ onto?" (Answer: Yes.) – fleablood Apr 19 '21 at 03:21
  • @fleablood I copied and pasted the question from the practice solutions so I did not add my own wording. Anyhow, I took the exam today making sure to practice by the book since other classmates were also complaining that there were at least 3 other wrong answers to the solutions from the professor… – Kevin Li Apr 20 '21 at 02:09
  • If your professor claimed at any time that $f:\mathbb Z \to \mathbb Z; f(x)= 2x-3$ is onto and did not have a typo or a slip of a tongue, then you have a fairly serious issue. That $f$ if not onto and it's pretty basic that it is not. – fleablood Apr 20 '21 at 03:45

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You can solve this by trying to calculate an (right-)inverse.

We have $f(m)=2m-3$. Now we set $x=2y-3$ and solve this for $y$.

This gives $y=\frac{x+3}{2}$, so the (right-)inverse would have to look like this:

$g:\mathbb{Z}\to\mathbb{Z}, m\mapsto \frac{m+3}{2}$, which is obviously not well definied, as it maps only onto $\mathbb{Z}$ if $m+3$ is even. And this holds only for odd integers. With other words, $m+3$ has to be even so that the experssion $\frac{m+3}{2}$ makes sense in $\mathbb{Z}$.

So every odd integer is an image in $\mathbb{Z}$ but not the even integers are not.

Cornman
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  • I’m fairly new to inputting mathematical notation, btw. Is there a markup language I need to learn to do things like the y=(x+3)/2? – Kevin Li Apr 19 '21 at 01:26
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    It is called MathJax, or LaTeX. You can look it up, or maybe press edit on my post, to see the code. It is fairly easy to pick up, as most expressions have intuitiv names. You can learn it on-demand from several websites, and look for the signs you are missing. – Cornman Apr 19 '21 at 01:29
  • I just realized I can copy and paste LaTeX from Microsoft Word’s equation editor… guess I’ll hold off learning the mark-up for a while. – Kevin Li Apr 19 '21 at 01:39
  • I would not advice that. Learning LaTeX is pretty simple. It might seem overwhelming at first, but you will get quickly, as it is so intuitiv. At first you should just know how to write fractions, and some special sets like $\mathbb{Z}$, then you will naturally improve over the time. A nice way to find symbols quick, is to draw them: https://detexify.kirelabs.org/classify.html – Cornman Apr 19 '21 at 01:51
  • Our meta site has it's own mathjax guide, and I would advise learning it. It's simpler than general LaTeX. – Lee Mosher Apr 19 '21 at 03:10