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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4
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3answers
191 views

Proving the area of a square and the required axioms

I recently realized the area formula of all polygons, and most basic figures can be proven from the areas of square and rectangle. For example if we know the area of rectangle, we can the area formula ...
1
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4answers
53 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 $1$?
1
vote
3answers
74 views

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 whats going on. An explanation would be nice.
3
votes
2answers
175 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
87 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
32 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
47 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
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1answer
44 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
42 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
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1answer
34 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
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0answers
133 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
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0answers
40 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
83 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
0answers
53 views

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

Let G be a group, g an element of G, and n a positive integer. 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 ...
5
votes
1answer
33 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
92 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
223 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 ...
0
votes
0answers
75 views

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

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
97 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 ...
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0answers
136 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
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0answers
39 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
73 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
110 views

Tricks - Prove Homomorphism Maps Identity to Identity - Fraleigh p. 128, 129 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'. Then $ \phi(a) = \phi(a\color{magenta}{e}) = ...
4
votes
1answer
29 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
34 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
115 views

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

(32.) Let H be a subgroup of a 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
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0answers
31 views

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

Let H be a subgroup of a group G and let $a, b \in G.$ Prove the statement or give a counterexample. If $aH= bH,$ (30.) then $Ha= Hb.$ (33.) then $a^2 H = b^2 H.$ I understand p. 3: Let $G = S_3$ ...
0
votes
3answers
57 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
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3answers
111 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
26 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).
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4answers
93 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 \\ ...
0
votes
2answers
67 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
60 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
47 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
37 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
145 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
56 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
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0answers
40 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)} ...
6
votes
0answers
47 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
46 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
63 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
370 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
99 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
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 + ...
0
votes
2answers
94 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
147 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 : ...
1
vote
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)} ...
1
vote
2answers
106 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
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 ...
14
votes
6answers
365 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 ...