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.

learn more… | top users | synonyms

0
votes
4answers
65 views

Let x and y be integers, let x and y be greater than 0. Prove that the gcd (x/gcd(x,y) , y/gcd(x,y) = 1

Very confusing, not really sure how I'm supposed to deduce what $\gcd (x,y)$ is and how $$\gcd \left(\frac{x}{\gcd(x,y)} , \frac{y}{\gcd(x,y)}\right)$$ can be $1$?
0
votes
3answers
78 views

Suppose $X$ and $Y$ are greater than $0$. Show that $\gcd(X,Y)$ is $1$ iff $\gcd(X^m,Y^m)= 1$

Problem Suppose $X$ and $Y$ are greater than $0$. Show that $\gcd(X,Y)$ is $1$ iff $\gcd(X^m,Y^m)= 1$. Please help with the above. I have no idea what's going on. An explanation would be nice.
3
votes
2answers
348 views

Fallacious Induction Proof that All Integers are Equal [duplicate]

We're currently learning about induction in my real analysis course. I came up with the following proof that is obviously false but cannot quite figure out why it fails. Here it is... "Any set $S$ of ...
2
votes
1answer
143 views

Injection function and product of two exponential elements - homomorphisms -

[Fraleigh, p.133, ex. 13.7] Let $f_i: G_i \rightarrow G_1 \times G_2 \times \dots \times G_r$ be given by $f_i(g_i) = (e_1, e_2, ..., g_i, ..., e_r),$ where $g_i \in G_i$ and $e_j$ is the ...
0
votes
2answers
36 views

Groups Math Proof Help

Show that the indicated set $G$ with the specified operation forms a group by showing that the four axioms in the definition of a group are satisfied. $G = \mathbb Z_5$ under addition mod $5$. I ...
4
votes
1answer
87 views

Image of Group Homomorphism is Finite and Divides |Domain of Group| - Fraleigh p. 135 13.44

Let $\phi: G \rightarrow G'$ be a homomorphism. Show that if $|G|$ is finite, then $|\phi[G]|$ is finite and divides $|G|$. Because $φ[G] = \{φ(g) \, | \, g ∈ G\}$, we see $|φ[G]| ≤ \quad |G|$ which ...
0
votes
1answer
74 views

Greatest Common Divisor Proof

If $d = \gcd(a,n)$, must $\dfrac ad$ and $n$ be relatively prime? Prove or disprove. Do I show that they need to be relatively prime and then the inverse that they do not need to be relatively ...
1
vote
1answer
54 views

Prove that for any two real numbers a and b $\big||a|-|b|\big|< |a-b|$ [duplicate]

I know I should use the triangle inequality.
1
vote
1answer
37 views

REF(A + B) = REF(A) + REF(B) [Strang P130 3.3.5]

Describe all $m$ by $n$ matrices $A$ and $B$ such that $ref(A) + ref(B) = ref(A + B)$. Is it true that $ref(A) = A$ and $ref(B) = B$? Does $ref(A - B) = rref(A - B)$? Here, ref = Row Echelon ...
1
vote
0answers
1k views

Easier Solution? - Find plane perpendicular to another plane and through the intersection line of two planes [Stewart P803 12.5.38]

$38.$ Find an equation of the plane that's $\perp$ the plane $x + y - 2z = 1$ and passes through the line of intersection of the planes $x - z = 1$ and $y + 2z = 3$. $\bbox[3px,border:2px solid ...
4
votes
0answers
53 views

Nontrivial homomorphism for $Z_a \times Z_b $to $Z_c \times Z_d$ - Fraleigh p. 134 13.35

This isn't a duplicate of this. Let $(A, B) \in \mathbb{Z_a \times Z_b}$. Hinging on p. 2, I guess homomorphism is $h(A,B) = (A \text{ mod } c, B \text{ mod } d)$. I'm unsettled. p. 2 sprang it up ...
3
votes
1answer
102 views

$K_{1,3}$ packing in a triangulated planar graph

I am trying to show that every planar triangulated graph $G=(V,E)$ with $|V| \ge 5$ has an edge decomposition into $|V| - 2$ groups of $K_{1,3}$. In other words, that we can pack $|V| - 2$ instances ...
6
votes
1answer
104 views

Necessary and Sufficient Condition for $\phi(i) = g^i$ as a homomorphism - Fraleigh p. 135 13.55

Let $g \in \text{ group } G $ and $n \in N$. Let $\phi : \mathbb{Z_n} \rightarrow G$ be defined by $\phi(i) = g^i$ for $0 \le i \le n$. Give a necessary and sufficient condition (in terms of g and n) ...
5
votes
1answer
38 views

Is there a nontrivial homomorphism for each of the given groups? - Fraleigh p. 134 13.38, 13.41, 13.43

(38.) $\mathbb{Z} \rightarrow S_3$? Let $φ(n) = \begin{cases} \mathrm{id} \in S_3 &, \text{for all $n$ even,} \\ \mathrm{transposition} (1,2) &, \text{for all $n$ odd integers.} ...
5
votes
1answer
128 views

Intuition - Homomorphic Image of Group Element is Coset - Fraleigh p. 135 13.52, p.130 Theorem 13.15

Theorem 13.15: Let $\phi: G \rightarrow G'$ be a group homomorphism, $g \in G$. Then $g\ker\phi = (\ker\phi)g = \operatorname{Im}^{-1} \left[ \; \{ \; \phi(g) \; \} \; \right] = \phi^{-1}[ \; \{ ...
3
votes
2answers
230 views

How to ensure that you haven't run into a paradox proving a theorem e.g. by proof by contradiction?

While preparing some lecture notes for next semester and going back to basics (set theory and proof strategies) I came along the following simple question which is about proving theorems in general ...
2
votes
4answers
120 views

Show Pascal triangle properties

I need to prove two pascal triangle properties: 1) $\sum_{k=0}^{n}\binom{p+k}{k}=\binom{p+n+1}{n}$ 2) $\sum_{k=0}^{n}\binom{k}{p}=\binom{n+1}{p+1}$ I need some advice on how to approach to this ...
1
vote
0answers
160 views

Choosing the vector that minimizes this sum related to the rearrangement inequality

The rearrangement inequality states that, for two sets of real numbers $x_1\leq\dots{}\leq x_n$ and $y_1\leq\dots{}\leq y_n$, the sum $\sum_{i=1}^n x_{\sigma(i)}y_i$ is minimized for the particular ...
1
vote
0answers
106 views

How to prove that the inverse of a persymmetric matrix is also persymmetric?

An exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...
2
votes
1answer
243 views

Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.

I am confused as to how to solve this question. For the Base case $n=1$, $(2^{2(1)} - 1)\,/\, 3 = 1$, base case holds My induction hypothesis is: Assume $2^{2k} -1$ is divisible by $3$ when $k$ is a ...
5
votes
3answers
259 views

Tricks - Prove Homomorphism Maps Identity to Identity - Fraleigh p. 128 Theorem 13.12(1.)

Let $\phi$ be a homomorphism of a group G into a group G'. If $e =$ the identity element in G, then $\phi(e) =$ the identity element in G'. Is this what Sharkos is trying to answer: About ...
4
votes
1answer
40 views

Intuition and Strategy - Index of Subgroup of Subgroup Proof - Fraleigh p. 103 10.38

This isn't a duplicate. I tried kb's answer and Answerer 1 but I'm still confounded. I like $\frac {\left| G\right| } {\left| H\right| }$ better than $[G:H]$ hence I write it as a fraction. Suppose ...
2
votes
2answers
36 views

Ordered abelian groups

Consider the following axioms: 1) $\ x+(y+z)=(x+y)+z$ ; $\forall x \forall y \forall z$ 2) $\ x+0=x$ ; $\forall x$ 3) $\forall x$ $ \exists y$ such that $\ x+y=0$ 4) $ \ x+y=y+x$ ...
6
votes
1answer
222 views

A subgroup has the same number of left and right cosets - Tricks - Fraleigh p. 103 10.32, 35

(32.) Let $H \le$ group G and let $a, b \in G.$ Prove or disprove. If ${aH= bH},$ then $Ha^{-1} = Hb^{-1}.$ $\color{blue}{Ha^{−1}} = \{\color{magenta}ha^{−1} | h ∈ H\} = ...
4
votes
0answers
48 views

Counterexamples to Nonidentities - Power of Cosets and Right Coset - Fraleigh p. 103 10.30, 33

Let $H \le$ group G and $a, b \in G.$ Prove or give a counterexample. If $aH= bH,$ (30.) then $Ha= Hb.$ (33.) then $a^2 H = b^2 H.$ I understand p. 3: Let $G = S_3$ and $H = \{(1), (1,3)\}$. ...
0
votes
3answers
76 views

Fixed point and period of continuous function

Prove/ Disprove: Let $f:(0,1)\to(0,1)$ be such that $|f(x)-f(y)|\leq 0.5|x-y|$ for all $x ,y.$ Then f has a fixed point. 2.Let $f:\mathbb R\to\mathbb R$ be continuous and periodic with period ...
1
vote
3answers
147 views

Construction of an osculating circle

Let $\alpha$ be a unit-speed curve.Then there exists a unique circle $\beta$ such that $\beta(0)=\alpha(0), \ \beta'(0)=\alpha'(0), \ \beta''(0)=\alpha''(0).$ Attempt: Consider $\beta(s)= \textbf{p} ...
0
votes
1answer
37 views

Prove If hcf(a,b)|c then then ax+by=c has an integer solution. Where a and b are non-zero integers.

I'm not sure whether to use multiple cases for this particular question (i.e. odd*odd with hcf=1 and odd*even with hcf=1 have integer solutions for x and y).
1
vote
4answers
95 views

Calculating $\dfrac{1}{2^x}$ using $5^x$

If we look at the decimal equivilents of $2^{-n}$, we see they resemble $5^n$ with a decimal point in front of them: $\begin{align} 2^{-1} &= 0.5 \\ 2^{-2} &= 0.25 \\ 2^{-3} &= 0.125 \\ ...
-1
votes
2answers
79 views

Show that $z$ is prime if $z|xy$ implies $z|x$ or $z|y$

Let $z$ be an integer greater than or equal to $2$. Suppose for all integers $x$ and $y$ that $z|xy$ implies $z|x$ or $z|y$. Show that $z$ is prime.
3
votes
2answers
67 views

Let $x$ be greater than $1$. Prove $x$ is prime if and only if for every integer $y$, either $\gcd(x,y)=1$ or $x\mid y$.

I've been having serious trouble with this problem, The first direction-> Proving x is prime if for every integer y, either gcd(x,y)=1 or x|y doesn't seem too difficult. We know that if gcd(x,y)=1 ...
0
votes
1answer
67 views

What does “possible to define” mean?

What does "possible to define" mean in general? First I thought it means that "can not lead to a contradiction", but such seems to be hard to prove. Then for the proof I was looking at, involving ...
3
votes
0answers
50 views

Using an automatic tool for checking geometric conjectures

I do a lot of research about squares, and I thought of using some automatic tool for proving / disproving some geometric conjectures. As a simple example, consider the following Square coloring ...
0
votes
1answer
152 views

Verification and help to simplify an argument about closure of some sets.

Hi everyone I'd like to know if what I have so far is correct, I think is much work for something which is too simple I would appreciate any advice or whatever. Moreover, I have doubt in (3) and (4), ...
0
votes
4answers
148 views

Harmonic number induction proof clarification

Number 8 in this solution set: http://www.cs.ucdavis.edu/~bai/ECS20/hw5sol.pdf How does the summation $\displaystyle \frac{1}{2^k+1} + \cdots \frac{1}{2^k+2^k}$ become $\displaystyle ...
1
vote
0answers
104 views

Proof-finding: Power iteration and complexity of the Rayleigh quotient

I'm searching for a proof for this theorem: \begin{align} |\lambda^{(k)}-\lambda_1| = \mathcal{O}\Big(\Big|\frac{\lambda_2}{\lambda_1}\Big|^{2k}\Big) \end{align} where \begin{align} \lambda^{(k)} ...
7
votes
0answers
75 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
49 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
71 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
1k 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
116 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
70 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 + ...
2
votes
2answers
132 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
258 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
199 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)} ...
2
votes
2answers
265 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
138 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
631 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
77 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 ...