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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0
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1answer
624 views

Double Complement of a set proof

Question states: Prove the law of double complements for sets: If $A$ is a set and $A^\complement$ is its complement than prove that: $$ (A^\complement)^\complement = A$$ I started with: $$ ...
2
votes
1answer
37 views

Formal proof structure for $\forall n \in \mathbb{N}, P(n) \rightarrow \forall n \in \mathbb{N}, Q(n)$

I'm used to proving universal quantification claims (i.e. $\forall n \in \mathbb{N}, [P(n) \rightarrow Q(n)]$) by: Assuming an arbitrary number in the naturals, assuming the antecdent $P(n)$, doing ...
3
votes
2answers
87 views

If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?

The title pretty much says it all: If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true? Edit: Let me attempt to be a little more precise: ...
1
vote
2answers
84 views

how to prove $\sum_{i=1}^n i^k =\Theta(n^{k+1})$

we can say that if all $i$ s in the sum were equal to $n$ then the answer to the summation would be $n\cdot n^k$. So $n^{k+1}$ is the upper bound.so $\displaystyle\sum_{i=1}^n i^k=O(n^{k+1})$ For ...
1
vote
1answer
60 views

Determining if two statements are equivalent, logical sense.

I am confused, I am working with proofs and I have the following statement to work with $\forall n\in\mathbb{N},P(n) \implies P(n+1)$ I have a second statement $\forall n\in\mathbb{N}, ...
2
votes
4answers
63 views

Prove $\frac{n}{n+1}<\frac{n+1}{n+2}$

How can we prove the following inequality: $$\frac{n}{n+1} < \frac{n+1}{n+2}$$ I understand how to do proof by inductions and contradictions, but I am getting stuck on this question. My proof ...
1
vote
1answer
37 views

Suppose n is an integer. Use a proof by contrapositive to show if n^3 is even, then n is even

So, we assume that n is not even. Then, $n$ is odd, so $n= 2k+1$ for some integer $k$. Then, $(2k+1)^3 = 8x^3+12k^2+6k+1$. Would it be legal, then, for me to say $(8k^3+12k^2+6k)+1 = ...
-1
votes
2answers
40 views

Suppose that x is an integer. Use a proof by contrapositive to prove that if 5x+7 is even, then x is odd.

I know that we assume x is even. So, as x is even, x = 2k for some integer k. Then, that would make for 5(2k)+7 = 10k + 7. And this is where I'm stuck. I know that it isn't complete at 10k+7 to ...
2
votes
1answer
116 views

Statistics - Show that $\hat{\theta}$ hat is a biased estimator of $\theta$

I'm asked to solve this exercise, but I can't manage to find something satisfying. Any help/hint would be much appreciated. Let $Y_1, Y_2,\dots, Y_n$ denote a random variable sample of size n from a ...
0
votes
1answer
50 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
votes
3answers
98 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
votes
1answer
22 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
votes
2answers
38 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 $f(x) = f(x')$. ...
0
votes
1answer
66 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
votes
3answers
81 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
35 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
votes
4answers
51 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
60 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
votes
3answers
32 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 ...
2
votes
2answers
53 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
28 views

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

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
vote
0answers
31 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 ...
7
votes
2answers
118 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
vote
1answer
25 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 $ac \gt bd$ ?
1
vote
0answers
46 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
votes
1answer
39 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
31 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
vote
1answer
49 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
941 views

Proof by induction that $3^n \geq 2n^2 + 3n$ for $n \ge 4$

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 ...
0
votes
2answers
79 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
374 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
votes
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
votes
2answers
44 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, ...
0
votes
2answers
48 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
votes
1answer
9 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 ...
1
vote
2answers
67 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
1answer
40 views

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

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

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

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
3answers
243 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
5k 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
46 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
22 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
75 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
vote
2answers
35 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
89 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
192 views

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 ...
2
votes
5answers
260 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
46 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
75 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 ...
1
vote
2answers
47 views

Showing the summation of numbers

Using each of the digits 1 through 9 once, form numbers whose sum is 100. If you think it can't be done, then prove it. My attempt: I say it can't be done because the sum of all numbers $1-9$ is ...