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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.

15776 questions
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Proof without using induction that a number is divisible by 6

Prove without using induction that all numbers of the form $6|8^n - 2^n$. I need a brush up on subtracting numbers with the same base but different exponent. So far I have $8^n - 2^n = 2^{3n} - 2^{n}$. Am I headed in the right direction?
mylasthope
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For every $x \in [\frac{\pi}{2},\pi]$, $\sin(x)+\cos(x)\geq 1$. Prove rigorously by contradiction.

For every $x \in [o,\frac{\pi}{2}]$, $\sin(x)+\cos(x)\geq 1$. How do you prove this rigorously by contradiction? I understand you start by assuming that $\sin(x)+ \cos(x)<1$ and prove this is a false statement. I can see from drawing the graphs this…
Tk706
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Prove that if a|c and b|c, and a and b are relatively prime, than ab|c

How do I show this? I have an idea of what to do, but the problem overall is a little confusing to me. I can start the problem, but I just do not see how to get to the solution.
JCMcRae
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Let n ∈ ℕ. If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

I know how to test the divisibility of a number by 9, but only if I am given what n is. How would I set this problem up?
JCMcRae
  • 843
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Euclidean algorithm to provde gcd's and multiples

Suppose $a, b, n ∈ \Bbb{N}$. Use the Euclidean algorithm to prove that $\gcd(na, nb) = n \gcd(a, b)$. I was going to try setting it up, by literally doing: $nb = rna + k$ and so forth, but something tells me this is wrong.
Test
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Use the Fundamental Theorem of Arithmetic to prove that if a>1 is composite, then there exists a prime p such that p|a and p≤√a

I know that since $a>1$ is composite, then it can be broken down into a product of prime factors, by Fundamental Theorem of Arithmetic. So then $a=p_1p_2\dots p_k$ for some natural number k. Then, I notice that since $a=p_1p_2\dots p_k$, then there…
JCMcRae
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If a theorem says "$A \iff B$" and I want to prove $A$, does it suffice to show $A \implies B$?

For example, if there is theorem that says: "$[x] = [y] \iff x \sim y$," and I am asked to prove $[(a,b)] = [(c,d)]$ Is it enough to show that $[(a,b)] = [(c,d)] \implies (a,b) \sim (c,d)$, because of the theorem that says "$[x] = [y] \iff x \sim…
mr eyeglasses
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rewrite $(\sqrt{5} + 2)(\sqrt{5}-2) = 1 $ until...

"rewrite $(\sqrt{5} + 2)(\sqrt{5}-2) = 1 $ until you have an equation with $\sqrt{5}$ on the left and a ratio of two expressions involving $\sqrt{5}$ on the right." Ok..All i need to know is if i'm reading this question wrong. Are they asking me if…
user125535
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Introduction chapter Exercise Q3 from "How to Prove It: A Structured Approach"

The following question is from the book "How to Prove It: A Structured Approach" Second Edition. Theorem 3 : There are infinitely many prime numbers. Euclid's proof Introduction Chapter : Exercise Question 3 The proof of Theorem 3 gives a method for…
Faizal Ismaeel
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Does this part of an arithmetic progression have a name?

In this arithmetic progression - 11+30w, 11 is the initial term, 30 is the common difference, and w is what? I use the letter w because it is the first letter of the word whole, and I use w to represent the whole numbers. So, 30·0=0. 11+0=11. When…
Jeffrey Young
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What is the proof of $n^2 = 1 + 3 + 5 ... (2*n - 1)$

What is the proof of $n^2 = 1 + 3 + 5 + ... + (2\times n - 1)$? While I verified that this is true for small numbers, I am looking for a mathematical proof for all Natural Numbers .
user3105485
  • 700
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Proof with Cartesian coordinates.

Let $S_b := \{(x,y) \in\mathbb R^2 | y = 3x + b\}$ where $b\in\mathbb R$. Give a direct proof that if $(r,s)\in\mathbb R^2$, then there exists a $b\in\mathbb R$ such that $(r,s) \in S_b$. I have not worked a proof with Cartesian Coordinates…
JCMcRae
  • 843
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Cartesian product proof with counterexample

I was asked to disprove the following statement by counterexample: Let A, B and C be sets. If A x C = B x C then A = B I was under the impression that: (x1, y1) = (x2, y2) if and only if x1 = x2 and y1 = y2. So with this definition I can't really…
jn025
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Prove there's always a larger element

We have a number $ 0 < x < 1 $. We also have the function $1-\dfrac{1}{n}$ with $n \in \mathbb{N}$. How can I prove that for any $x \in \mathbb{R}$, there exists an $n \in \mathbb{N}$ such that $ 1-\dfrac{1}{n} > x$? Of course, my intuitive problem…
skaf
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Prove by contraposition, if n is a positive integer such that n(mod3)=2 then n is not a perfect square

the question: Prove by contraposition, if $n$ is a positive integer such that $n(\mod 3)=2$ then $n$ is not a perfect square. I've started by negating the statement, "Not q then not P": Suppose if n is a perfect square, then n is not…
I hate math
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