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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63 views

What is wrong with the proof given below?

This problem comes from Solow's book, 2nd edition. What is wrong with the proof given below? If $r$ is a real number with $|r| \leq 1$, then for all integers $n \geq 1, 1 + r + r^2 + \ldots + ...
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2answers
90 views

Does dividing by zero ever make sense? [duplicate]

Good afternoon, The square root of $-1$, AKA $i$, seemed a crazy number allowing contradictions as $1=-1$ by the usual rules of the real numbers. However, it proved to be useful and ...
3
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3answers
140 views

All possible values of $i^{-2i}$ - NBHM $2013$

Question is to write down all possible values of $i^{-2i}$ I know that $e^{i\theta}=\cos(\theta)+i\sin (\theta)$ So, I can write $i=e^{i.\frac{\pi}{2}}$ then I would have : ...
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1answer
36 views

Proof regarding limit with 2 variables

I've encountered a problem which I would like some assistance in doing. Determine the values of $p$ for which the following limit does or does not exist: $$\lim_{(x,y)\to (0,0)} ...
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2answers
103 views

Basic proof problem from “How to Prove it A Structured Approach”

I got the book How to Prove it A Structured Approach and I'm ashamed to admit I failed to even do the first problem in the introduction chapter: a) Factor $2^{15} - 1 = 32767$ into a product of two ...
2
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1answer
128 views

Analytic continuation of zeta is meromorphic on $\mathbb{C}$ with simple pole at 1

We have the following identity: For some contour $\gamma$ and $\forall s \in \mathbb{C} $ Re $s > 1$: $$-2i\sin(\pi s) \Gamma(s)\zeta(s)= \Large\int_{\gamma} \frac{(-z)^{s-1}}{e^z-1}dz$$ The ...
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6answers
362 views

When to use the contrapositive to prove a statment

My question tries to address the intuition or situations when using the contrapositive to prove a mathematical statement is an adequate attempt. Whenever we have a mathematical statement of the form ...
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1answer
71 views

Simple trick in a integral

I am studying a proof of a theorem and in the proof i have the following equality $$ \int_{0}^{\pi /2} \frac{\left|\sin(2nt)\right|}{t} \ \mathrm dt= \sum_{k=1}^{n} \int_{0}^{\pi} \frac{\sin(t)}{ t ...
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2answers
65 views

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 ...
2
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4answers
125 views

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 ...
2
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1answer
78 views

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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2answers
38 views

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 ...
2
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4answers
55 views

$E$ 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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1answer
34 views

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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1answer
85 views

Proof by contradiction: May I assume $P$ (true) in $\neg Q \land P \Rightarrow P \land \neg P$ to prove $Q$ by contradiction

Suppose I wish to do a proof by contradiction the statement $Q$. In proving $Q$ may I assume $\neg Q \land P$ (where $P$ is a statement known to be true) and show the implication $\neg Q \land P ...
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1answer
232 views

Intuition - If $Ax = b$ has infinitely many solutions, why can't $Ax = c$ have only one solution? [Strang P165 3.4. 22]

If $\mathbf{Ax = b}$ has infinitely many solutions, why is it impossible for $\mathbf{Ax = c}$ (where $\mathbf{c}$ is a new right side) to have only one solution? Proof : Take two solutions of ...
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1answer
31 views

Collection of Intuitive / Visual Derivations of Mathematical Concepts and Formulas

I find it difficult to simply memorize mathematical formulas in engineering without understanding what it means and what the result is like, but I realized that many mathematical relationships can be ...
3
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1answer
73 views

Partition the following collection of groups into subcollections of isomorphic groups - Fraleigh p. 84 8.10

Here a * superscript means all nonzero elements of the set. The orange is the answer. Then S = $\{C_1, ..., C_9\}$ is a partition of the given collection into subcollections of isomorphic groups. ...
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0answers
112 views

Are these sets with orbits subgroups of $S_A$ = set of all permutations of A? - Fraleigh p. 86 8.41-8.43

In Exercises 40 through 43, let $A$ be a set, $B$ a subset of $A$, and let $b$ be one particular element of $B$. Determine whether the given set is sure to be a subgroup of $S_A$ under the induced ...
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3answers
98 views

Show that for any sets $A,B$ and $C$ $A\Delta B\subset A\Delta C\cup B\Delta C$.

The problem statement is in the title. I'm proving a problem in class and it's necessary for me to show the above containment. I've drawn some Venn diagrams to make sure the containment actually ...
2
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1answer
30 views

Basic inequality proof construction

Given: $x \le y$ and $z \le 0$ Use the result "if $x \le y$, then $-x \ge -y$" To prove that $zy \le zx$
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2answers
98 views

How to show that $x$ becomes a root of $p(x)$ in $F[x]/(p(x))$

$F$ is a field, $p(x)$ is irreducible polynomial at $F[X]$. $K=F[X]/\left<p(x)\right>$. For every $a\in F$ we will mark: $\bar{a}=\left<p(x)\right>+a$. Now, the question is: How do I show ...
0
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1answer
38 views

Prove that if $a\neq b$ and if $(x-a)$ & $(x-b)$ divide $f(x)$, then $ (x-a)(x-b) \mid f(x)$

Prove that if $a\neq b$ and if $(x-a)$ & $(x-b)$ divide $f(x)$, then $(x-a)(x-b) \mid f(x)$ We know that $(x-a)\mid f(x)$ & $(x-a) \mid f(x)$, but what does this implies for polynomials ...
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2answers
53 views

Induction proof equivalence

In Induction, we do the following: Check $P(1)$ is true, then show that if $P(k)$ is true, then $P(k+1)$ is also true. So we proceed to assuming $P(k)$ is true, then attempt to show $P(k+1)$ is true, ...
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3answers
96 views

General Topology: Neighborhood

Show that a subset $U$ of the real numbers is open in the usual topology if and only if, for all $x$ in $U$, there is a number $\epsilon>0$ such that $|y-x|<\epsilon$ implies $y$ is in $U$. "I ...
1
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2answers
60 views

How to prove $\sum_{k\leq n}^{n} \binom{n}{k}= 2^n$ by induction [duplicate]

$\sum_{k\leq n}^{n} \binom{n}{k}= 2^n , n, k \in \mathbb{N}$ Im trying with mathematical induction but im stuck. My inductive step: $H) \sum_{k=0}^{h} \binom{h}{k}= 2^h$ $T) \sum_{k=0}^{h+1} ...
0
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0answers
30 views

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 ...
0
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1answer
24 views

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 ...
1
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1answer
65 views

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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5answers
144 views

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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2answers
128 views

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 ...
0
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1answer
63 views

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 ...
3
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1answer
84 views

Induction Proofs in Abstract Algebra

In several abstract algebra textbooks, I have been seeing propositions that I would think require induction verified without using induction. For example, consider the claim that if $G_{1}, \ldots, ...
2
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1answer
84 views

A proof of an interesting Geometric Vector Theorem.

Suppose $O$ is the centre of the circumscribing circle of triangle $ABC$ and $H$ is its orthocentre. Prove that vector $OH$ is equal to the sum of the vectors $OA$, $OB$ and $OC$. An answer I ...
3
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1answer
43 views

How to prove this vector direction angle identity?

How do I go about proving this: $\cos⁡^2α+\cos⁡^2β+\cos⁡^2γ=1$? It's so different from normal trig proof, because the angles are not the same and everything is $\cos$. What steps should I take to ...
0
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2answers
87 views

Why $I=\left\{p(x)\in \mathbb{Z}\left[X\right]:2\mid p(0)\right\}$ is not a principal ideal? [duplicate]

I saw this question but I still do not understand: What is the difference between ideal and principal ideal? At my homework I had to prove to things about $I=\left\{p(x)\in ...
3
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1answer
162 views

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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0answers
123 views

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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1answer
25 views

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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1answer
95 views

Proof e is irrational — Choice of Sign of m/n = e and Uncanny Step

Assume $e$ is rational. Then, there exist coprime integers $m$ and $n$, and we can choose $n$ to be positive, such that: $\displaystyle \frac m n = e = \sum_{i \mathop = 0}^\infty \frac 1 {i!}$ from ...
4
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2answers
103 views

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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4answers
243 views

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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3answers
75 views

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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2answers
98 views

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 $3k+2$. ...
4
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1answer
255 views

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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4answers
78 views

Proof strategy to show $\large \sum_{0 \ \le \ x \ \le \ a} \normalsize \binom{a}{x} \binom{b}{n+x} = \binom{a+b}{a+n} $

The following two combinatorial identities are taken from a textbook. $$\begin{align} &\large \sum_{0\ \le \ x \ \le \ n} \normalsize \binom{a}{x} \binom{b}{n-x} = \binom{a+b}{n} \tag{10} \\ ...
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3answers
77 views

For a finite field $F$ of order $n$, all elements are roots of $x^n - x$

I need to prove two things at $F[X]$ but don't know how, Ill glad to get help... $F$ is a finite field. $|F|=n$. We look at $p(x)=x^n-x\in F[X]$ 1. How we can show that every $c\in F$ is a root of ...
1
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1answer
30 views

Is the approach to proving this expression for an n-choose-k algorithm correct?

I randomly encountered this post here, asking why this is true: ...
5
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3answers
61 views

Do Question's Given GCD Statements Imply these New GCD Statements?

Are the following statements true or false, where $a$ and $b$ are positive integers and $p$ is prime? In each case, give a proof or a counterexample: (b) If $\gcd(a,p^2)=p$ and ...
0
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2answers
83 views

General Proof that $\left< v, v \right> = \left|\left| v \right|\right|^2$

Consider $\mathbb{R}^n$ with the standard Euclidean inner-product. I'm trying to give a proof that $$ \left< v, v \right> = \left|\left| v \right|\right|^2 $$ but can only seem to do it for ...