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I am still learning about mathematical proofs and wanted to ask for some feedback, if possible, of my solution for the following problem.

Let $x,y\in \mathbb{Z}$. Prove that $x-y$ is even if and only if $x$ and $y$ are of the same parity.

To prove this I used a Lemma and the following theorem:

Theorem 16 Let $x,y\in \mathbb{Z}$. Then $x$ and $y$ are of the same parity if and only if $x+y$ is even.

My approach was the following:

Lemma If $n\in \mathbb{Z}$, then $n$ and $-n$ are of the same parity.
To prove this lemma we consider two cases.
Case 1. Assume $n$ is even. Then $n=2k$ for some $k\in \mathbb{Z}$. Therefore, $-n=-(2k)=2(-k)$. Since $-k$ is an integer, it follows that $n$ is even.
Case 2. Let $n$ be odd. Then $n=2k+1$ for some $k\in \mathbb{Z}$. Thus, $-n=-(2k+1)=-2k-1=-2k-2+1=2(-k-1)+1$. Since $-k-1$ is an integer, $-n$ is odd.

We now proceed to prove the result. Let $x$ and $y$ be of opposite parity. By lemma, $x+(-y)$ is a sum of two integers of opposite parity. Therefore, by theorem 16, $x-y$ is odd.
For the converse, assume $x$ and $y$ are of the same parity. By lemma, $x+(-y)$ is a sum of two integers of the same parity. Therefore, by theorem 16, $x-y$ is even.

Thank you for your attention :).

Kr'aamkh
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    It looks fine to me. – José Carlos Santos Sep 21 '21 at 17:57
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    It looks like you started from the axioms and then proceeded your way to a proof which one of many ways to do a proof. From what I am told, this is called a synthetic proof. Check out this article. It really helps: https://deopurkar.github.io/teaching/algebra1/cheng.pdf – James Sep 21 '21 at 18:03
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    The question "Can this be considered a formal proof?" has an answer. That answer is "no". A formal proof is basically a proof so precise and which so rigidly follows formal rules that a machine could check the proof to verify it's correct. That doesn't take anything away from your proof, which is perfectly fine and rigorous. – Mark Saving Sep 21 '21 at 18:18
  • Thanks for your feedback and links for information!! Sorry for using the term "formal proof" in an incorrect way (I had no idea it was a specific term lol). I think what I meant was "rigorous proof" (and this was clarified by the feedback from everyone). Despite of that, I surely learned something new which is always welcome. – Kr'aamkh Sep 22 '21 at 00:01

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This is an elaboration on my comment:

The question "Can this be considered a formal proof?" has an answer. That answer is "no". A formal proof is basically a proof so precise and which so rigidly follows formal rules that a machine could check the proof to verify it's correct. That doesn't take anything away from your proof, which is perfectly fine and rigorous.

A formal proof must take place in a specified formal system. Some examples of formal systems are sequent calculus, natural deduction, and Hilbert-style rules.

There are also variants of dependent type theory which can be used as formal systems, though these are not commonly taught or used in undergraduate mathematics.

Your proof is "informal" in the sense that although it is a correct proof, it is not a formal proof in any of the above styles. It can, however, be translated into a formal proof in an obvious way. It is therefore rigorous.

Mark Saving
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