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 views

Characterization of analytic functions by exponential functions

Let $f$ be an analytic function on domain $D$ such that $f(z) \neq 0, \forall z \in D.$ Could anyone advise me how to prove $f= e^{h},$ for some analytic $h$ on $D \ ?$ Thank you.
0
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0answers
13 views

How to prove that $f(n) + O(f(n)) = \Theta(f(n))$?

I am trying to prove that $f(n) + O(f(n)) = \Theta(f(n))$ Here's what I've done so far: $f(n) = \Theta(f(n))$ $C_1 f(n) < f(n) < C_2 f(n)$ $C_1f(n) + O(f(n)) < f(n) + O(f(n)) < ...
0
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3answers
42 views

Proofs about Matrix Rank

I'm trying to prove the following two statements. I can prove them easily by considering the matrix as a representation of a linear map with a given basis, but I don't know a proof which uses just the ...
0
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1answer
16 views

Help understanding proof for: Let $X$ be a set. Then $X \not\approx P(X)$ (where $\approx$ is equivalence relation)

In trying to understand the following proof, I am getting stuck on the chosen definition of $Y = \{ x \in X \mid x \not\in f(x) \}$. How do we know that such a set exists in $P(X)$ when we don't even ...
3
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2answers
28 views

We are given $f: X \rightarrow P(X)$, $f(x) = X\backslash\{x\}$, and $X$ is a set. Is the function injective, surjective, bijective?

I am working on this problem in a beginners set theory class. I believe the function is injective but not surjective, thus is it not bijective. We can show it is injective by letting $f(x) = f(x')$. ...
-1
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0answers
29 views

trigonometric equation (proof answer) [on hold]

hi,all as you can see in the picture there are two parts that need to be proof. first is based on (b) and second based on (a) for the first equation, i already got the answer which is d3=2dm2. ...
0
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1answer
18 views

Field Proofs with Multiplicative Inverses

I've been staring at these two for a while and I can't come up with anything concrete to start. Hints on how to begin would be greatly appreciated, full solutions are not necessary (and preferably ...
0
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0answers
43 views

Proving that $3 = 9^{-1} \pmod{26}$

Prove that $3$ is the multiplicative inverse of $9 \pmod {26}$ $$\quad26\quad1\quad0\\2\quad9\quad0\quad1\\\;\;1\quad8\quad1\quad{-2}\\\quad\;1\quad-1\quad3$$ Hence $3$ is the multiplicative inverse ...
0
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3answers
18 views

Game Theory Voting Utilities

! So far, I've managed to come up with this solution: ! But as far as here...I can convert this into payoffs, however I'm unsure of how to figure out the Nash equilibria as when we convert from ...
1
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0answers
13 views

Prove that a mixed strategy in two player, zero sum, matrix game must exist (alternative proof)

So I am having a trouble with this game theory proof. I feel pretty good with my answer for part 1, but I am not really sure how to get started on the rest of it. Any help would be appreciated. Let ...
0
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4answers
44 views

Prove that for all integers a and b, a + b and a − b are either both odd or both even.

Stumped on this proof. I've only been able to figure it out assuming that both a and b are even: $a = 2k$ and $b = 2n$ $2k + 2n = 2(k + n)$, definitely even. $2k - 2n = 2(k - n)$, also definitely ...
1
vote
2answers
34 views

Proving by induction, if the base case fails to meet the main condition, what do we do?

I have to determine the number $x$ of subsets with odd cardinalities of a set $S$ and then prove that I'm correct. I determined the number $x$ is obtained using the formula $2^{n-1}$ where $|S| = n$. ...
0
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3answers
17 views

Bounded Sequences and Extrapolation of Convergence From Related Sequences

I'm considering some sequence $S_n$ which is bounded, and I want to prove that $S_n/n$ is convergent. I'm thinking that I could simply take $lim_{n \to \infty} S_n/n$ and simplify this to $(lim_{n \to ...
0
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0answers
26 views

Proving that a value is a multiple of 11

I need to prove that 12**n - 1 is a multiple of 11 for every value of n (part of N). This is clearly a proof-by-induction problem. My base case is 0, where I assume n = 0 will give a resulting 0 ...
0
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0answers
13 views

How to prove the deduction theorem in a natural deduction calculus?

In van Dalen's Logic and Structure, after defining the notion of a derivation (p.35-6) and syntactic consequence (p.36) the author immediately exhibits some lemmas (1.4.3) where the deduction theorem, ...
2
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2answers
39 views

Showing that $(\mathbb{R} \setminus \{ 0 \}, \, \times) \not \cong (\mathbb{C} \setminus \{ 0 \}, \, \times)$

I'm trying to show that $(\mathbb{R} \setminus \{ 0 \}, \, \times) \not \cong (\mathbb{C} \setminus \{ 0 \}, \, \times)$ as follows: note that there exists an element (namely $i$) in $\mathbb{C} ...
1
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0answers
20 views

Prove the congruence $pB_{p-1} \equiv -1 \pmod p$ for Bernoulli numbers.

I need to prove that: If $p$ is prime greater than or equal to five, then $pB_{p-1}$ belongs to the p-integers and more over: $$pB_{p-1} \equiv -1 \pmod p$$ Hint:Put $N=p$ in the Faulhaber´s ...
1
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0answers
27 views

How to prove this

Prove that if $A^x+B^y=C^z$ where $A,B,C,x,y,z$ are positive integers and $x,y,z$ are all greater then $2$ then $A,B,C$ must have a common prime factor. I heard about this problem a long time ago. I ...
6
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2answers
93 views

$\gcd(a,b)$ compared to $\gcd(3a,b)$

$\gcd(a,b)=\gcd(3a,b)$? They are obviously not equal in general, as $\gcd(ax, bx)=|x|\gcd(a,b)$.
1
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1answer
21 views

Inequalities with variables that are integers

If $a, b, c, d$ are all positive integers, is it true that if $a \gt b$, and $c \gt d$, then we can say that $ac \gt bd$ ?
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0answers
21 views

Prove the von Staudt-Clausen congruence of the Bernoulli numbers

I need to prove that: If $p$ is prime greater than or equal to five, then $pB_{p-1}$ belongs to the p-integers and more over: $$pB_{p-1} \equiv -1 \pmod p$$ Hint:Put $N=p$ in the Faulhaber´s ...
0
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1answer
32 views

Prove that $M_{p}$ is an ideal of the p-integers

I need to prove that: $M_{p}:=\{ x \in \mathbb{Q}:|x|_{p}<1\}=\{ \frac{a}{b} \in \mathbb{Q}:b\in \mathbb{Z}-p\mathbb{Z},a \in p\mathbb{Z} \}$ Is an ideal of the p-integers and p-integers/ $M_{p}$ ...
0
votes
2answers
24 views

Limit of |x-2| as x approaches -2

I believe that it equals -4. In the epsilon-delta definition, we can set delta equal epsilon and I become this satisfies the definition. The problem is I can't seem to prove based on this that 0 less ...
1
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1answer
23 views

Prove some properties of the $p$-adic norm

I need to prove that the p-adic norm is an absolut value in the rational numbers, by an absolut value in a field I mean a function that goes from $K \to \mathbb{R}_{\ge 0}$ such that: I)$|x|=0 ...
4
votes
5answers
306 views

Proof by induction involving inequalities

Problem: If $n$ is a natural number and $n\geq4$, then $3^n \geq 2n^2 + 3n$. (Prove by Induction.) Attempt at solution: 1) Given: $n$ is a natural number, $n \geq 4$. 2) Let $P(n)$ be the statement ...
-3
votes
0answers
22 views

Is it possible to prove that this Hessian matrix is greater than $0$? [on hold]

Is it possible to prove that this Hessian matrix is greater than $0$? Given $f(0)=g(0)=0$, $f'>0$, $f''<0$, $g'>0$, $g''<0$, and if $c'(n)<0$ then $f'(\theta) - \alpha<0$, and ...
-5
votes
2answers
54 views

If $x \in \mathbb{R}$ and $x \neq 15$, then $x^3 - 5x^2 + 3x \neq 15$ [on hold]

I tried this and managed to disprove it. Not sure if this is correct. Someone please help me
0
votes
2answers
49 views

Pair of positive integers in product sums

I am still not sure on this answer. I would like someone to help me see the solution to his question. I was working on it for a while and it is the only question that I looked at that I can not ...
0
votes
4answers
41 views

How to write a formal proof of the statement: For all integers n, if n is a multiple of 5 then 3n is a multiple of 5.

Prove: For all integers $n$, if $n$ is a multiple of $5$ then $3n$ is a multiple of $5$. Proof: Assume $n$ is a multiple of $5$. Then $n$ must have the form $5k$ where $k \in \mathbb{Z}$. We have ...
0
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0answers
21 views

Distance between points in the plane [duplicate]

I have this problem and I honestly don't even have a clue of how to start, would someone help me please? Let $A$ = {$v_1$,$v_2$, . . . ,$v_n$} be a set of points in the plane such that the distance ...
0
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2answers
40 views

Determine where do a function has limit.

I have to do the next exercise: Define $f:\mathbb{R} \to \mathbb{R}$ as follows: $$f(x)=x-[x]$$ if $[x]$ is even, and $$f(x)=x-[x+1]$$ if $[x]$ is odd. Determine those points where $f$ has a limit, ...
-2
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0answers
32 views

Prove that $f(x)= \frac{2x^{2}+3x-2}{x+2}$ has limit at (-2) and other exercises. [on hold]

I have to prove the next statements: 1)Define $f:(-2,0) \to \mathbb{R}$ by $f(x)= \frac{2x^{2}+3x-2}{x+2}$.Prove that $f$ has a limit at -2, and find it. 2)Suppose $f:D \to \mathbb{R}$ has limit at ...
0
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2answers
27 views

Direct proof involving ceil and floor - Homework

The exercise Proof the following directly: Let $x \in \mathbb{R}$. If $x \in \mathbb{Z}$, then $\lfloor x \rfloor = \lceil x \rceil$. My problem I mostly fail completely on the formal part of ...
0
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1answer
7 views

Given two tangents $\varepsilon_{1},\varepsilon_{2}$ of the curve $c_{1}$, on two specific points $x_1, x_2$, find the tangents

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
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2answers
43 views

Sufficient condition for $f(z)$ to be polynomial

I think it suffices to exhibit a sequence $\{R_n\}$ of positive real numbers such that $R_n \to \infty$ with $f(z) \neq 0,$ whenever $|z|=R_n$ and $\begin{align} ...
0
votes
0answers
31 views

Verbal Traps with Numbers vs Percents [on hold]

While problems of cohesion are acknowledged, there is evidence to suggest there is much that unites people from different ethnic groups. Regardless of ethnicity, people strongly identify with ...
0
votes
1answer
33 views

Proof By Induction that $3^{(2^n)} -1$ is divisible by $2^{(n+2)}$ [on hold]

How do I prove the $(n+1)$-th case for this equation?
2
votes
1answer
44 views

Proving $2^{\mathbb N}$ is uncountable [on hold]

I know that there is a way to prove this and I am trying to figure this out using Cantor's diagonal method. For the most part, I understand how it works somewhat. You basically try to prove that ...
1
vote
2answers
82 views

Can the proof of Theorem 1.20 (b) in the book, The Principles of Mathematical Analysis by Walter Rudin, 3rd ed., be improved?

I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, third edition, and am at Theorem 1.20(b), where he states and proves that between any to real numbers, there is a rational; that is, if ...
0
votes
6answers
120 views

Prove that the square root of 3 is irrational [duplicate]

I'm trying to do this proof by contradiction. I know I have to use a lemma to establish that if $x$ is divisible by $3$, then $x^2$ is divisible by $3$. The lemma is the easy part. Any thoughts? How ...
0
votes
1answer
22 views

How to prove that the L2 norm is a non-increasing function of time for a 2nd-order PDE?

I am having a test in few days and I saw an interesting question while I was skimming through the book problems. The problem is concerned about initial-boundary value problem of 2nd order PDEs. To ...
0
votes
0answers
9 views

Tips for proving biconditionals involving ORs

I'm attempting to prove a statement, but am not sure of the formal method for proving this kind of construction. So, I am having trouble starting off the proof. The statement is constructed in the ...
0
votes
1answer
47 views

An entire function with an integral bound for $f'/f$ on a sequence of circles must be a polynomial

Let $f(z)$ be entire. Suppose there exists $M >0$ and sequence $\{R_n\}$ of positive real number tending to $\infty$ such that $f(z) \neq 0$ and $|z|=R_n,$ such that $\begin{align} \int_{|z|=R_n} ...
1
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2answers
20 views

Show: $\max_{|z|=R} \operatorname{Re}\left(z\frac{f'(z)}{f(z)}\right) \geq N $

Let $f$ be a holomorphic function defined in a neighbourhood of $\overline{D(0,R)}$ which has no zero on $\partial D(0,R).$ Let $N$ be number of zeros of $f$ in $D(0,R).$ Show: $\max_{|z|=R} ...
2
votes
1answer
70 views

Determine if there is an integer $n\geq 1$ such that $(\sqrt{2}+1)^{1/n}+(\sqrt{2}-1)^{1/n}\in\mathbb{Q}$

could you help me with this problem? I have to determine whether there exists a number n larger or equal to 1 for which the number $$ \sqrt[n]{\sqrt{2}+1}+\sqrt[n]{\sqrt{2}-1} $$ is rational. I still ...
5
votes
0answers
150 views
+50

How to show that this function respects the strict ordering of its input.

Suppose you have a vector $\pmb x=\{x_i\}_{i=1}^n$ where each entry is drawn from a continuous distribution and $n$ is even. Then, denote $i^*=\{1\leq j\neq i\leq ...
0
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0answers
72 views

Let Cn denote the number of ways of writing a valid list of open and closed parentheses of length 2n

(a) Let Cn denote the number of ways of writing a valid list of open and closed parentheses of length 2n (valid means that at any point along the list, the number of open parentheses must be greater ...
2
votes
5answers
218 views

Cosine of the sum of two solutions of trigonometric equation $a\cos \theta + b\sin \theta = c$

Question: If $\alpha$ and $\beta$ are the solutions of $a\cos \theta + b\sin \theta = c$, then show that: $$\cos (\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2}$$ No idea how to even approach the ...
0
votes
1answer
19 views

Proof limit of ratio of sequence .

Prove that as $n\to\infty$ $$\frac{1}{x_n} \to \frac{1}{x}$$ where we are also given $x_n \to x$, and $x_n,x\neq0$ Attempt: Suppose $x_n \to x$. Then for every $\epsilon > 0$, there exists a ...
0
votes
1answer
20 views

Square root of Sequence approaches square root value.

Suppose that $x$ is a real number, and $x_n\geq 0$, and $x_n→x$ as n grows. Prove that $\sqrt {x_n}→\sqrt x$ as $n$ grows. Attempt: Case 1: $x = 0$. Suppose that $x$ is a real number, and $x_n \geq ...