# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### Did I go about determining the coplanarity of these three vectors wrong?

I asked this question a few days ago, where the question was this: I have a task stating this: Determine if the following vectors are coplanar. Assume that $v_1$, $v_2$ and $v_3$ are ...
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### Proof that $x^k < k^x$

So, I want to prove that $x^k$ is less than $k^x$ for any $x > k$. $x$ and $k$ are both integers. My first approach was an induction over $k$, given that the numbers are integers. I also ...
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### Philosophical side of MATH. knowing the path then walk it. [closed]

Can I find a book that gives me the purpose of theorems and definitions without going deep into proofs. It's just like knowing the path then walk it. That's will me the understanding reach the next ...
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### Second isomorphism theorem for subspaces

just like I did some days ago, I now have to show that $T/T\cap U \cong (U+T)/U$. Therefore I tried finding a surjective homomorphism and then, by using the first isomorphism theorem, I should be ...
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### $E$ Closed iff $\partial E \subseteq E$

I'm having trouble verifying my proof, would appreciate some input on this one. Let $(X,d)$ be a metric space with $E\subset X$. Suppose $E$ is closed in $X$, which means that $E=\overline{E}$. By ...
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### What matrix corresponds to the sum of the column space of two matrices? [Strang P131 3.1.31]

$P124:$ The column space consists of all linear combinations of the columns. The combinations are all possible vectors $\mathbf{Ax}$ and fill $C(A)$. The columns of $A$ and $B$ and $M$ are all ...
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### A function $f: H \to \mathbb{R}$ is not weakly continuous at $0$ but $(f(x_n)$ converges to $0$ whenever $(x_n) \to 0$ weakly in $H$

Let $H$ be a Hilbert space equipped with its weak topology and let $K \subset H$ such that $K = \{ \sqrt{n}e_n | n \in \mathbb{N_0} \}$ Let $f:H \to \mathbb{R}$ be a function such that $f(x) = 1$ when ...
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### Given $a, b, c, d, m \in\mathbb{Z}$such that $5\mid (am^3 + bm^2 + cm + d)$, prove that there exists integer $n$ such that…

Given $a, b, c, d, m$ in $\mathbb{Z}$ such that $5|(am^3 + bm^2 + cm + d)$ and $5 \not| d$ , prove that there exists an integer $n$ such that $5\mid(dn^3 + cn^2 + bn + a)$ I've spent about two hours ...
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### General topology: Showing a set is open.

I am using Munkres and the problem states, let $\textbf{X}$ be a topological space: let $\textbf{A}$ be a subset of $\textbf{X}$. Suppose that for each $x \in \textbf{A}$ there is n open set ...
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### Are $p \to (q \to r)$ and $p \to (q \wedge r)$ logically equivalent?

Is $p \to (q \to r)$ logically equivalent to $p \to (q \wedge r)$? I simplified each one, I got $\neg\, p \vee(q \vee r)$ and $\neg\, p ∨(\neg\, q \wedge r)$ respectively. Not sure if my ...
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### How many digits do we need for a proof ??

In the question: Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx$, the value of that integral was ...
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### Prove meromorphic function can be written as product of holomorphic and rational function

I'm not able to prove this. Any help would be welcomed ! Let U be a simply connected domain and let $f$ be a meromorphic function on U with only finitely many zeroes and poles. Prove that there is ...
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### If two sets have the same cardinality, then so do their power sets. Converse can't be answered?

For infinite sets $A, B$, $|A| = |B| \Longrightarrow \require{cancel} \cancel{\Longleftarrow} |P(A)| = |P(B)|$. I recast http://ph.answers.yahoo.com/question/index?qid=20100907061641AAE2Vfq : ...
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### Questions on Answer to “The cardinality of the set of all finite subsets of an infinite set”

Would someone please enlarge on Arturo Magidin's original answer ? $1.$ Say the question didn't divulge $|S| = |X|$. Then how can $|S|$ be determined? Any intuition? I recast it below with more ...
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### A problem on path-wise connectedness

Let $K = \{(x,y) \in \mathbb{R^2}| x=0,-1 \leq y \leq 1\}$, $G=\{(x,y) \in \mathbb{R^2}| \ 0<x \leq 1, y=\text{sin}(\frac{1}{x})\}$ and $A=K\bigcup G.$ Claim: $A$ is not pathwise connected. ...
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### A Householder matrix is symmetric

I want to show that a Householder matrix is symmetric, so I must show that $H^T = H$, but from the formula $$H= I - (uu^T/\beta),$$ they are not equal. What's wrong with my reasoning? EDIT: I ...
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### For which primes p is $p^2 + 2$ also prime?

Origin — Elementary Number Theory — Jones — p35 — Exercise 2.17 — Only for $p = 3$. If $p \neq 3$ then $p = 3q ± 1$ for some integer $q$, so $p^2 + 2 = 9q^2 ± 6q + 3$ is divisible by $3$, ...
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### Intuition — An integer $n > 1$ is composite $\iff \color{purple}{p \le \sqrt{n}}$ divides it.

Origin — Elementary Number Theory — Jones — p32 — Lemma 2.14 Backward direction — I need to prove there exists a divisor $d$ of $n$ satisfying $1<d<n$. Because $p$ is prime, $1 < p$. ...
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### Proof - There're infinitely many primes of the form 3k + 2 — origin of $3q_1..q_n + 2$

Origin — Elementary Number Theory — Jones — p28 — Exercise 2.6 To instigate a contradiction, postulate $q_1,q_2,\dots,q_n$ as all the primes $\neq 2 (=$ the only even prime) of the form $3k+2$. ...
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### Proof — Infinitely many primes of the form $4k + 3$ — origin of $4(p_1…p_k - 1) + 3$

I know there are sundry questions — like this pdf — and this (10.) Prove that any positive integer of the form $4k + 3$ must have a prime factor of the same form. Because $4k + 3 = 2(2k + 1) + 1$, ...
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