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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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Direct Proof Rational numbers

Prove that for each positive integer $r$, there exists a rational number $c$, s.t. $c > r.$ Could we let $c$ = $(r+1)$$/1$ and then conclude $r+1 > r$ so $c > r$?
Jay3
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Format for proving more than two statements are equivalent

When proving say $P \iff Q$, we generally write the proof in this fashion: Proof: $\Rightarrow$ [Proof that $P$ implies $Q$] $\Leftarrow$ [Proof that $Q$ implies $P$] QED. Now, say we wanted to prove $P_1\iff P_2\iff P_3 \iff P_4$. We would show…
Bonnaduck
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If $xy > 0$ then $x > 0$ and $y > 0$ or $x < 0$ and $y < 0$.

I'm not sure how to prove that if $xy > 0$ then $x > 0$ and $y > 0$ or $x < 0$ and $y < 0$ just by using the ordered field axioms. Can someone perhaps help me see this through?
lithium123
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Simple example of a Heuristic proof method

of all the proof methods I had never seen the Heuristic proof method before, I came across this http://oddperfect.org/pomerance.html , seemed too complicated to be used for understanding the method itself, proof by contradiction uses irrationality…
jimjim
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How to describe "the digits of n" mathematically where n is an integer?

Suppose n = 12345 The sum of the digits of n = 1 + 2 + 3 + 4 + 5 = 15 For example, in Python, we might isolate the digit 1 by writing n[0]. How would one represent the digits of n mathematically?
sammy
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How to construct a proof once we intuit a solution

For any integer N, there is an integer P such that one of the following is true: N = 10P N = 10P + 1 N = 10P + 2 N = 10P + 3 N = 10P + 4 N = 10P + 5 N = 10P + 6 N = 10P + 7 N = 10P + 8 N = 10P + 9 We know that there are 9 integers between every…
sammy
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How to write this proof in a purely formal way?

Problem. Let $\Lambda$ be an index set. Then show that $$\displaystyle\bigcup_{\alpha\in\Lambda}\left(\displaystyle\bigcup_{B\in\gamma_{\alpha}}B\right)=\displaystyle\bigcup_{B\in\gamma}B$$where…
user170039
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Did I prove and disprove the following statements correctly?

Let $A = \left\{x \in \mathbb{Z} \mid \exists a\in\mathbb{Z}: x = 6a + 4\right\}$ and $B = \left\{y \in \mathbb{Z} \mid \exists b\in\mathbb{Z}: y = 18b - 2\right\}$ and $C = \left\{z \in\mathbb{Z} \mid \exists c\in\mathbb{Z}:z = 18c +…
antman1p
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How to do proofs involving sets

I have just recently started preparing for a course I will be taking next year, but I have very limited knowledge as it relates to proofs. It seems as though the only proofs I am slightly familiar with are proofs by induction. I was wondering if…
HASSOUN
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Prove $\forall n \in\mathbb{N}, (n|105 \wedge n|70) \implies 5|n$

I know the definition of divides into is $$a|b \equiv \exists a\in\mathbb{Z}, b = ac$$ however I'm not sure how to manipulate this to prove $$\forall n \in\mathbb{N}, (n|105 \wedge n|70) \implies 5|n$$ Any help to lead me in the right way would be…
LAJ
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How to prove a function from A to B

I have a question that says... THEOREM: The function $f: \mathbb{R}_{\geq 0}\rightarrow \mathbb{R}$ given by $f(x) = ln(x)$ is onto. If you were going to prove this statement, what is the first sentence. After this sentence what is the new goal? So…
user6259845
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Proof for statement: It's impossible to find a pair of consecutive natural numbers whom digit sums would divide without reminder by 3

I am searching for a mathematical proof of this statement: It's impossible to find a pair of consecutive natural numbers whom digit sums would divide without reminder by 3. I have tried: To make a system of linear equations where $\overline{xy} =…
Zyberg
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Does proof proceed from left to right?

Very simple question: if we're asked to prove that $a=b$, do we start with $a$ and then find $b$ from $a$? Does going the other way around count as a formal proof? The exercise in question is this: Prove that $$\mbox{arctanh}\left[ \frac{1}{\cosh…
inspd
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Need help with proofs using axioms only

Prove if $p,q∈R$ and $pq>0$ then either $p>0$ and $q>0$, or, $p<0$ and $q<0$ using only the field axioms. I have no idea how to do this using only the field axioms. Seems pretty straightforward but how would you approach this question using only the…
Mathew
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(trivial proof) show that $(x-1)(x-3)\geq -2$

I'm taking my first math course that requires me to write proofs, and even though I understand most of the course material, I'm struggling with actually proving things in a rigorous way. For example, we're asked to show (using what they've called…
Lincoln77
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