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.

15776 questions
8
votes
2 answers

How does proof by contradiction work?

I just had a thought from looking at this answer in this thread: For every $\epsilon >0$ , if $a+\epsilon >b$ , can we conclude $a>b$? I understand Brian's proof, but I just thought of something. If you prove a hypothesis is untrue, then the…
Twenty-six colours
  • 1,881
8
votes
2 answers

True for all $n$ implies true as $n$ tends to $\infty$?

I'm doing some exercises and came across one that has two parts, as follows: Given a transition matrix for a Markov Chain, $\mathbf{P}$, and a vector $\mathbf{f}$, $\mathbf{f}$ is harmonic if $$ \mathbf{f} = \mathbf{P}\mathbf{f}$$ $(a)$ Show that if…
Wilduck
  • 183
8
votes
3 answers

Difference between proof and plausible argument.

I wanted to ask about the difference between a proof and a plausible argument. What is difference in proving a statement and providing a plausibility argument for it. The example on which I am basing it is: We know that in numerical analysis, the…
kusur
  • 585
8
votes
3 answers

How do I know when to use "let" and "suppose" in a proof?

When the goal is $∀n\in\Bbb N ∀m\in\Bbb N (n \ge m \rightarrow H_n-H_m \ge {n-m \over n})$, I can begin the proof with "Let n and m be arbitrary. Suppose n and m are natural numbers" or "Let n and m be arbitrary natural numbers." The boundary…
user231595
7
votes
0 answers

Prove that the only prime triple is 3, 5, 7

Prove that the only prime triple is $3,5,7$. I tried proving using this method: Multiplication of $3$ jumps back and forth between being an even and an odd number. Thus goes from odd to odd over an interval max size 6, and likewise from even to…
user1090614
  • 427
7
votes
6 answers

Prove that there are no $x,y ∈ \mathbb N$ for which $x^2-y^2 = 10$

I began by factoring and got $(x+y)(x-y) = 10$ Then I tried cases and was able to prove the ones where $x$ and $y$ are equal-> because the equation will result to zero. and also where $x < y$, because the answer will be negative. How can I prove…
Mac
  • 941
7
votes
5 answers

Prove that $r+x$ is irrational

If $r$ is rational ($r$$\ne$$0$) and $x$ is irrational, prove that $r+x$ is irrational. Assume that $r+x$ is rational. Then $r+x=(\frac{p}{q})$, where $p$ , $q$ are $\in$ $\mathbb{Z}$, and $p$ and $q$ are in lowest terms. Then we have…
Skm
  • 2,296
6
votes
3 answers

Prove that: $(1+a_1)(1+a_2)...(1+a_n) \le 1 + S_n + (S_n)^2/2! + ... + (S_n)^n/n!$

I am currently working on this problem from Hardy's Course of Pure Mathematics and have gotten stuck near the end. I was wondering if someone could help me determine what to go next. Question If $a_1, a_2, ...,a_n$ are all positive and…
GovEcon
  • 2,656
6
votes
4 answers

I've worked out the reasoning, but how do I write the proof?

This started our with a pretty trivial problem that went: Fill in the blanks with whole numbers to make mathematically true statements. Do not use the same number twice within a statement.$$\frac{*}4+\frac{1}*=\frac{*}{20}$$ Now solutions were…
Brendan
  • 131
6
votes
1 answer

The author of my book simplifies his solutions to an extent that I am uncomfortable with, so are my solutions to homework over doing it?

This question can be summarized as: How explicit does one need to be when writing proofs? To what extent can one implicity write a proof safely? The first chapter of our text in elementary discrete math is very brief in the solutions. As an example…
Leonardo
  • 988
6
votes
1 answer

Proof convexity of mean absolute error

I am given the cost function: mean absolute error $$ \frac{1}{N} \sum_{n=1}^{N} \big|y_n - f(x) \big|$$ If $f(x)$ is a linear regression: $f(x) = x^Tw$, where $w$ are the parameters of the regression function and $x$ the input, how can I prove that…
james
  • 303
6
votes
7 answers

prove: if n is even, then n+1 is not even

This proof seems so simple that it's hard (if that makes any sense.) based on the definition, n is even iff there exists k such that n = 2k. What I really want to say is (big picture) By definition, let $n = 2k.\;$ Then $n+1 = 2k + 1$. $2k + 1$ is…
electr0hed
  • 157
  • 1
  • 2
  • 5
5
votes
1 answer

Help getting started on a proof

I honestly have no idea where to start with the following proof, and I was wondering if anyone could help me get started. I don't want the whole idea, I just need to know where to start with this proof. The question: Let x be a real number. Suppose…
benty
  • 317
5
votes
2 answers

Prove! that $a+(1/a) ≥ 2$ and $a+(1/a) ≤2$

Let $a \in R$ If $a>0$, then $a+\frac1a\geq2$ If $a<0$, then $a+\frac1a\leq2$ This is how someone explained the first one to me but still not really sure about it. Proof: $\Longleftrightarrow$$a+\frac1a\geq2$ $\Longleftrightarrow$ the square of…
SpankyS
  • 71
5
votes
1 answer

Why is $(a+b)^3=a^3+3a^2b+3ab^2+b^3$ not a mathematical statement?

STATEMENT: $$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$ As per my knowledge, a sentence is said to be a mathematical statement iff it fulfills both of the following criteria: The sentence should be declarative. It should have a definite truth value. The…
CalIMpire257
  • 61
1
2
3
…
50 51