Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

Questions with this tag are about the presentation of a mathematical proof. Questions might include:

  • Should I include [x-mathematical detail] at [y-part of this proof]?
  • Is the following a sufficient proof of [x-mathematical tidbit]?
  • I have written the following proof, could I somehow improve it, does it have good flow/can I improve readability?

But this tag is not for asking someone else to write a proof for you, or for how to answer some question. Questions such as: My professor asked me to prove the Pythagorean theorem and I don't know how to begin are not to have this tag.

This tag is intended for use along with other, more "mathematical" tags. A question about the writing of a proof in abstract algebra, for example, should have as well. This tag can be used along with the proof verification tag.

See here for a useful set of guidelines for writing a solution.

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Is there a "rule of thumb" of what can be reasonably omitted from a proof at the graduate student level?

As someone entering graduate school this fall, this is something I would like to know. For undergraduates, I know that professors generally want students to show very rigorously and clearly their chain of reason when writing proofs. This generally…
Joseph DiNatale
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Homework: prove there are two distinct integers $m$,$n$ such that $1/m+1/n$ is an integer.

I know this is Nth duplicate of this question, as I've seen it before, and I've been reading some of the answers as well. However, being really a total beginning with proofs, I'd appreciate if someone could comment on the following. The task is to…
2305843008139952128
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Prove that there does not exist a surjective function from the set of rationals to reals.

Prove that there does not exist a surjective function f: $\mathbb{Q}\rightarrow \mathbb{R} $. I think a proof by contradiction would work which means we want to prove $$\neg (\forall{y}\in \mathbb{R}, \exists{x}\in\mathbb{Q}, ~f(x) = y)$$ which is…
Daniel Cazares
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help on proof writing: do I need to explicitly state this trivial assumption?

Prove that $\sqrt{5}$ is an irrational number. Part of the answer: Let $x^2=5$ and $x=p/q$ where $p$ and $q$ are integer numbers and $\operatorname{hcf}(p,q)=1$. $$\begin{align*} x^2&=5\\\ \left(\frac{p}{q}\right)^2&=5\\\ \frac{p^2}{q^2}&=5 &…
John
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What is wrong with this proof by contradiction?

My professor tells me that a proof by contradiction structure below is poor: Proof. To prove: A implies B. So, assume A (in order to prove that B follows). Assume not B (in order to arrive at a contradiction). Succeed at showing that not A follows…
John Davies
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Prove that $n + 2$ is odd where $n = 2k+1$ for some integer $k$

I am attempting to learn about mathematical proofs on my own and this is where I've started. I think I can prove this by induction. Something like: $n = 2k+1$ is odd by definition $n = 2k+1 + 2$ (this is where I'm stuck, how do I show that this is…
hereslookinateuclid
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What should be proven when your claim ends with a clause for a particular case?

My question is not about how to prove the following proof, as I already know how to do so. What I am more concerned about is the type of claim that is being made, and what am I really being asked to show; in particular when your claim is something…
Alex
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Is it proof by contradiction?

In a proof of the statement "A closed ball $\bar{B}\left(a,r\right)$ is a closed set", author first took complement of $\bar{B}\left(a,r\right)$ and proved it an open ball. I was wondering whether the author used proof by contradiction here.
Ansu
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injectivity of $f(x) = x^n$

In the first few lessons of an introductory Calculus class, there is this exercise: Show that the polynomial $f(x) = x^n$ is injective whenever $n$ is odd, and is not injective whenever $n$ is even. I understand why this statement holds…
Claire
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How do I write a formal proof related to this frog jumping puzzle.

There is a fairly well known puzzle about moving frogs that can be seen here. I found an interesting property about these puzzles and am certainly not the first to notice it. I can state this property informally, but I don't know how to formally…
user1153980
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Very basic question about the first step and goal in proof writing

I am new to proof writing. If trying to prove for example: $$(ab)^{-1} = a^{-1} b^{-1}$$ What am I trying to do - I either pick the left or right side as my starting point and manipulate it algebraically in isolation of the opposing side until I…
lopan
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What are some examples of proof by contrapositive?

Applying the Modus Tollens argument to Fermat's Little Theorem really helped me to understand logical implication. I never knew that FLT was actually a compositality test. Theorem (FLT): given integers $a>1$ and $n>1$, if $n$ is prime, then $a^n$ is…
cyclochaotic
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When to use For any vs Let at the start of a proof?

I was wondering when to use "for any"; vs "let...be..." at the start of a proof? For example, a proof may start like this: For any $a$ invertible in F, there exists $a^{-1}$ such that $ a^{-1}\cdot a =1. $ Then ..... vs. Let $a$ in F be an…
yi G
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Question of style: equiv vs. equals

Sometime I have trouble discerning whether an $=$ or an $\equiv$ is most appropriate. I believe that $\equiv$ is typically used when a new definition is being introduced, rather than a statement expressing a result. But even if that's essentially…
synaptik
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How to properly express that a proof of a proposition verifies a similar one simply by exchanging $\cap$s and $\cup$s?

I want to write a proof for two similar propositions, both proofs are similar except that all the $\cap$s in the first proof are replaced with $\cup$s respectively. How can I write formally and clearly, I mean, like a native-speaker, and avoid the…
闫嘉琦
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