# Tagged Questions

For questions about the formulation of a proof, not about the mathematics behind it.

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### Minimum spanning tree edge count

Given is a weighted complete graph where every weigth is a positive ineger. Let n be the amount of vertices. I have to prove that the number of edges of a minimum spanning tree of that graph is equal ...
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### Proving the closed form of $\sin48^\circ$

According to WA$$\sin48^\circ=\frac{1}{4}\sqrt{7-\sqrt5+\sqrt{6(5-\sqrt5)}}$$ What would I need to do in order to manually prove that this is true? I suspect the use of limits, but I don't know where ...
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### Logic - Hare and Tortoise Race.

Tortoise and Hare are racing. Tortoise moves one meter every minute. Hare moves $2n − 1$ meters on the nth minute (which is equivalent to $n^2$ meters in the first minutes). Prove that no matter ...
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### Good way to state, in english, the negation of “$f$ equals zero almost everywhere”.

Let $X = (X, \mathcal{E}, \mu)$ be a measure space and let $f$ be a measurable function on $X$. Consider the statement: $f$ equals zero almost everywhere Is there an concise, unambiguous way of ...
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### Let $A$ be a set and $f$ a function with $f:A\to A$

Here are the problems below. My main concern is with part a. I get confused when we at first define $f$ to make to $A\to A$, doesn't this imply that $f$ is onto already? Or is this just a bound given. ...
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### Prove $\forall x >0, \quad \sqrt{x +2} - \sqrt{x +1} \neq \sqrt{x +1}-\sqrt{x}$

I would like to prove $$\forall x >0, \quad \sqrt{x +2} - \sqrt{x +1} \neq \sqrt{x +1}-\sqrt{x}$$ I'm interested in more ways of proving it My thoughts: \begin{align} \sqrt{x+2}-\...
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### Prove $\lambda \notin sp(f) \iff \text{Ker}(f-\lambda Id)=\{0\}$

Let $E$ be a vector space. Let $f$ be an endomorphisme of $E$. Prove that $$\lambda \notin sp(f) \iff \text{Ker}(f-\lambda Id)=\{0\}$$ I'm interested in more ways of proving it We ...
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### Prove that if x is the square of any positive integer, then x mod 3 cannot equal 2

I tried to prove this by contradiction: Suppose $x = k^2$ for some positive integer $k$. Suppose $x\mod 3 = 2$ 2 = x - 3 $\lfloor x/3 \rfloor$ $n \le x/3 \lt n+1$ $3n \le x \lt 3n+3$ But then I'...
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I want to show you an example what my book says If $f:X\to Y$, $\mathfrak{C}$ is a collection of subsets of $X$, then $f(\bigcup \mathfrak{C})=\bigcup f(\mathfrak{C})$. Proof: (..) Let $y\... 1answer 34 views ### euclidian algorithm proof I am having difficulty with the following proof: The Euclidean algorithm can be used to express$x := gcd(a, b)$in the form$x = ma + nb$with$m, n \in Z$. Use this fact to prove the ... 2answers 78 views ### Trivialness of Center of Odd Dihedral Group [duplicate] So I'm trying to prove the center of Dn is trivial for odd n greater than or equal to 3. I know that since Dn for n greater than 2 is non-Abelian, the flips and the rotations do not commute in general.... 2answers 126 views ### Help to find all different cases need for proof about homomorphism from Z to R I am a bit confused about why my professor approached the following a certain way, and also why it cannot be done differently. The question is to prove that for any ring R we there is a unique ... 1answer 48 views ### Maximum sum of polynomial - choice of coefficient and variables There are two given arrays:$[x_1,...,x_n] $and$[y_1, ..., y_n]$. Our task is to make pair such that: $$\sum_{i=1}^{n} x_iy_j$$ is maximum. I know that we should sort these arrays:$x_1 \le ...\...
I suspect that $O(n)$ is homeomorphic to a product of spheres $S^m$ (equipped with the product metric) for various $m$ like so: $$O(n) \cong S^{n-1} \times S^{n-2} \times \dots \times S^0$$ I need ...