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 this a perfectly rigorous proof for "if and only if"? Any missing bits to achieve perfect rigor?

Note: my goal is maximum mathematical rigor. The toy problem here is (from [{Spivak's Calculus book}'s 1st chapter]'s problems): Prove that, if $b,d\ne 0$, then $\frac{a}{b} = \frac{c}{d}$ if and only if $ad = bc$. Allowed axioms are only: …
caveman
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Number of points on the elliptic curve $\ y^2 = x^3 + 1$

Consider the elliptic curve defined by $\ y^2 = x^3 + 1\ $ over $\ \mathbb{Z}_p,\ $ where $\ p \equiv 2 \pmod{3}\ $ is prime. Prove that the number of points on the curve is exactly $\ p + 1.\ $ Hint: for $\ y \in \mathbb{Z}_p,\ $ prove that there…
Apple
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Prove that $f(x)\leq x\cdot\log_2 x$ for all integer $x\geq1$

Let $f$ be a function which satisfies that $f(x) = \left \{ \begin{matrix} 0 & \mbox{for }x=1 \\ 2\cdot f(\frac{x}{2})+x, & \mbox{for }x\geq1 \end{matrix} \right.$ Prove that $f(x)\leq x\cdot\log_2 x$ for all integer $x\geq1$
Cicero Samuel
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Advice regarding preliminary proof of an untrue theorem

I am writing about objects A and B. They rely on well-known object C. I have defined A and B to be equal but I can use C to show that they are not equal. My intention is to show that C cannot be as commonly defined. Here is my question: Is it…
Jonathan Tooker
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Do all natural numbers exist on these three lines?

The lines given by $$f_1:\mathbb{W}\rightarrow \mathbb{N},~~f_1(n)=(6n+1)\cdot2^{k}$$ $$f_2:\mathbb{W}\rightarrow \mathbb{N},~~f_2(n)=(6n+3)\cdot2^{k}$$ $$f_3:\mathbb{W}\rightarrow \mathbb{N},~~f_3(n)=(6n+5)\cdot2^{k}$$ where $n,k$ are arbitrary…
Alex Maslach
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Proving If $S= \sum_i^n s_i >0$ then $S>s_i$ for some i

(assuming $n$ is finite) This seems like an easy proof, but how could one write it down nicely? I was thinking about proving it by cases: if one $s_i$ is negative, say $s_k$ than the statement is trivially true, with $S>s_k$, If all $s_i$ are…
user106860
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Right strategy to proof that $\forall b, c \in \mathbb{R}, b \gt 1: \exists n_0 \in \mathbb{N}: \forall n \ge n_0: n^c \le b^n$?

The true statement I fail to prove: $\forall b, c \in \mathbb{R}, b \gt 1: \exists n_0 \in \mathbb{N}: \forall n \ge n_0: n^c \le b^n$. Basically, as far as I can see, it says that every polynomial eventually grows slower than every $(> 1)$…
Zazaeil
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I don’t understand logic in proofs

I just had a general question about proofs. For “If A, then B” statements, we prove them by assuming the if statement is true and then find a way to get the consequent to be true. But for example the statement, “if 1=2, then 1+1=4” here the…
Paul Ash
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(0,n) can't map to (0,1) without dividing by some f(n).

I was trying to map a set x=(0,n) to y=(0,1) by using a single(non-piecewise) function. For example, you could say: f(x)={0, x=0 && 1, x \dne 0} however this is a piecewise function. I wanted the function to like this f(x) = x/x. However, this…
Jesse
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Prove that if $\{7k:k\in{\mathbb Z}\}\subsetneq\{nm:m\in{\mathbb Z}\}$, then $n=1.$

Let n be a natural number. Prove that if $\{7k:k\in{\mathbb Z}\}\subsetneq\{nm:m\in{\mathbb Z}\}$, then $n=1$. I know that we must show $x\in{A}$ implies $x\in{B}$, and that there exists $x\in{B}$ such that $x\notin{A}$, which means that…
macy
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Suppose the logical expression $(((\neg$ $P$) $\leftrightarrow$ $Q$) $\rightarrow$ $R$) $\vee$ ($P$ $\leftrightarrow$ $R$) is FALSE.

Let $P$, $Q$, and $R$ be statement variables. Suppose the logical expression $(((\neg$$P$) $\leftrightarrow$ $Q$) $\rightarrow$ $R$) $\vee$ ($P$ $\leftrightarrow$ $R$) is FALSE. What are the possible truth values for $P$, $Q$, and $R$?
macy
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Let $a$ and $b$ be two integers. Prove that if $ab=4$, then $(a-b)^3-9(a-b)=0$.

Let $a$ and $b$ be two integers. Prove that if $ab=4$, then $(a-b)^3-9(a-b)=0$. I have tried to approach this by proving the contrapositive instead, but I'm not sure if that's the best approach to this.
hunnybuns
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Prove the following: Let $n$ be an integer. If $2|n^2$, then $4|n^2$.

I came up with the following: $2|n^2$ implies that $2|n*n$. We proved in class that if $q|b*p$, then $q|b$ or $q|p$. Therefore, if $2|n^2$, then $2|n$ or $2|n$. So, $2|n$ implies $n=2k$, for some $k ∈ Z$. So, $n^2|(2k)^2$, $n^2 = 4k^2$, where…
Mettal
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proof with inequalities simple question

How do you prove: Suppose $x$ is a real number. if $x^3-x>0$ then $x>-1$ It seems really easy to do the contrapositive here i think but dont now how to word it. So suppose $x \le 1$ then $x^3 \le x$ for all x? would this be ok Thanks
Carlos Bacca
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proof on difference of two squares and odd integers

How would you prove that every odd integer is a difference of two squares? I re phrased the problem to make it clearer to me: If $k$ is an odd integer then it ca be expressed in the form $a^2-b^2$ where a and b are integers. So i start by supposing…
Carlos Bacca
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