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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7
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0answers
64 views

Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}$ isomorphic to the Klein $4$-group - Fraleigh p. 110 Exercise 11.12

I tried to fill in the steps but I'm still confounded by this solution. $|\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}| = 2\cdot2\cdot4$. $order(\mathbb{Z_2} \times \mathbb{Z_2} \times ...
0
votes
2answers
48 views

Prove that vectors x,y are linearly dependent exactly when …

Prove that vectors $\vec{x},\vec{y}$ (belonging to $\mathbb{R}^3$) are linearly dependent only if the following is true $$ \begin{vmatrix} x_1&y_1 \\ x_2&y_2 \end{vmatrix} ...
1
vote
3answers
70 views

Prove that $\mbox{Ker}(L)=\mbox{Ker}(L^2)$ if $\mbox{Im}(L) = \mbox{Im}(L^2)$

Let $L$ be a linear image from $\mathbb R^n$ to $\mathbb R^n$ that has $\mbox{Im}(L)=\mbox{Im}(L^2)$ Prove that $\mbox{Ker}(L) = \mbox{Ker}(L^2)$ I've been trying to get this for like two hours but ...
6
votes
1answer
686 views

Find all proper nontrivial subgroups of Z2 x Z2 x Z2 - Fraleigh p. 110 Exercise 11.10

$\newcommand{\lcm}[0]{\mathrm{lcm}}$I tried to fill in the steps but I'm confounded by this solution. Here $i$ is the identity element, not $e$. Because $\lcm(2, 2, 2) = 2$ hence all non-identity ...
0
votes
1answer
112 views

Does Graphical evicence count as / contribute to a Proof in Mathematics?

Several questions such as the following have an answer with pictures in it. How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$ How prove this inequality ...
1
vote
2answers
68 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 + ...
1
vote
2answers
122 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
votes
3answers
196 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 : ...
8
votes
1answer
186 views

The only element of $S_{\large{n \ge 3}}$ satisfying $\sigma y = y\sigma$ for all $y \in S_n$ is the identity permutation - Fraleigh p. 86 8.47

I don't want to type Greek letters hence I replaced $\gamma$ by $y$. Microsoft didn't replace them all. Call the identity permutation $id$. Prove the contraposition: $\sigma \neq id \implies ...
1
vote
1answer
37 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)} ...
1
vote
2answers
209 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
votes
1answer
135 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 ...
14
votes
6answers
469 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 ...
1
vote
1answer
75 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 ...
1
vote
2answers
85 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
votes
4answers
126 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
votes
1answer
83 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 ...
0
votes
2answers
54 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
votes
4answers
63 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 ...
1
vote
1answer
57 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 ...
0
votes
1answer
102 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 ...
3
votes
2answers
463 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 ...
1
vote
1answer
38 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 ...
4
votes
1answer
191 views

Partition groups into subcollections of isomorphic groups - Fraleigh p. 84 8.10 (please revamp title if necessary)

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. ...
5
votes
0answers
130 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 ...
2
votes
3answers
128 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
votes
1answer
32 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 $zy \le zx$
0
votes
2answers
101 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
votes
1answer
55 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 ...
1
vote
2answers
57 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, ...
2
votes
3answers
108 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
vote
2answers
63 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
votes
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
votes
1answer
27 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
vote
1answer
70 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 ...
4
votes
5answers
156 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 ...
1
vote
2answers
134 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
votes
1answer
93 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
votes
1answer
114 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
votes
1answer
113 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
votes
1answer
73 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
votes
2answers
92 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
votes
1answer
302 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 : ...
5
votes
0answers
171 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 ...
0
votes
1answer
29 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. ...
5
votes
2answers
264 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 ...
7
votes
4answers
269 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 $3$, ...
3
votes
5answers
202 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, $1 < p$. ...
5
votes
2answers
133 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
votes
1answer
716 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 $4k + 3 = 2(2k + 1) + 1$, ...