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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1answer
62 views

How to justify “two-dimensional” induction

Suppose I have a statement $p(m,n)$ where $m,n \in \mathbb{N}$ that I want to prove. Suppose I have proofs of the following: $p(1,n)$ is true for all $n \in \mathbb{N}$. $p(m,1)$ is true for all $m ...
2
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1answer
41 views

$4x^2+1$ factors only into $4y+1$ primes

How can one prove that numbers of the form $4x^2+1$ can only be divided by primes of the form $4y+1$ (e.g. there is no $x$ for which $7$ divides $4x^2+1$)? On a quick lookup, the statement is given ...
2
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1answer
35 views

Is $f: F \to R, \ (a_j)_{j \in \mathbb N} \mapsto \sum_{j \in \mathbb N} \ a_j $ bijective and find the inverse function!

$F$ is the set of the sequences in $\mathbb C$ and $R$ is the set of the series in $\mathbb C$. $f: F \to R, \ (a_j)_{j \in \mathbb N} \mapsto \sum_{j \in \mathbb N} \ a_j $ Now $\sum_{j \in \mathbb ...
0
votes
1answer
30 views

How to introduce bi-conditional in this proof?

This is from Discrete Mathematics and its Applications Just for context, I know that the universal set is everything and that the complement of A is just difference of the universal set and A. A ...
1
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0answers
41 views

Complex number - prove an inequality

Question: Given that:$$z^n\tan\theta_0 + z^{n-1}\tan\theta_1 + z^{n-2}\tan\theta_2 + ... + \tan\theta_n = 3$$ And that $\theta_i \in (0, \frac{\pi}{4})$, prove that: $$|z| > \frac{2}{3}$$ ...
2
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1answer
45 views

Complex numbers - minimum value proof

Question: For:$$|z - z_1|^2+|z - z_2|^2+|z - z_3|^2+\cdots+|z - z_n|^2 = S$$ Prove that the minimum value of $S$ is when:$$z = \frac{z_1+z_2+z_3+\cdots+z_n}{n}$$ I have no idea how to even ...
0
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1answer
49 views

Base 10 proof strategy

Let $a$ be a number written (in base 10) as $$a=a_0\cdot10^0+a_1\cdot10^1+a_2\cdot10^2+\cdots+a_n\cdot10^n$$ where $0\leq a_i <10$. Prove the following: 2 divides $a$ if and only if 2 ...
1
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2answers
73 views

How tot start proving $A \times B \times C \ne (A \times B) \times C$?

This is a problem from Discrete Mathematics and its Applications: Explain why $A \times B \times C$ and $(A \times B) \times C$ are not the same. I understand the process behind the ...
3
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2answers
120 views

Second Grade Homework Problem - Methodology

Ashamed to admit that I cannot aid my friend's niece with her second grade homework problem. So much for that college education, eh? Here's the problem. Using only the natural numbers 1 through 9 ...
0
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1answer
78 views

Property of continuous functions regarding maximum

Claim 1: If $f: [a, b] \rightarrow \mathbb{R}$ is continuous, then $f$ assumes a maximum value I know there's a theorem that states if $f$ is a continuous real-valued function on a closed interval ...
0
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1answer
62 views

Proof strategy for simple proofs.

I'm currently in a discrete mathematics course and I'm having quite a bit of trouble with the idea of proofs. From what I understand the one I've been stuck on is also rather simple but to me it's ...
3
votes
1answer
46 views

Using Mac Shane's Lemma

Let $I \subset \mathbb{R}^{N}$ be a convex, bounded open set with Lipschitz boundary $\partial I$. Let $\lbrace u_{n} \rbrace_{n}$ and $u$ be such that $$ u_{n} \rightharpoonup^{*} u~~ \text{ in }~ ...
4
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0answers
83 views

Is $\frac{\pi}{e}$ an algebraic integer?

From what I know, it is still an open question whether or not $\frac{\pi}{e}$ is irrational, but is there a proof that $\frac{\pi}{e}$ is not an algebraic integer?
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0answers
20 views

How do I prove that as 2 integers p, s tend to infinity, p/s tends to x?

Forgive me for asking such a broad question, but I really do have very little knowledge on how to do this and it came up in a problem that I have been working on for some time now, so any help would ...
0
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4answers
39 views

Proof that composition of invertible linear transformations is invertible (without determinants)

A crucial concept in linear algebra is that the composition of two invertible linear transformations is itself invertible. Here is the first proof I learned of this fact: Proof: Suppose that $T_1: ...
2
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2answers
75 views

Closed form solution and combinatorial proof.

First of all, I would like to figure out a closed form solution for the following summation: $$\sum^{n}_{k=0} C(n,k)\cdot C(2n,n+k)$$ Where C(n,k) means n choose k, or $\frac{n!}{(n-k)!\cdot k!}$ ...
2
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1answer
93 views

Suppose $R$ is partial order, prove that $R^{-1}$ is also a partial order

Suppose $R$ is partial order, prove that $R^{-1}$ is also a partial order. A partial order is a binary relation that is reflexive, anti-symmetric and transitive. So if $R$ is a partial order, ...
0
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1answer
53 views

Doubt : Invariance in Geometry

I was working my way through some Proof Problems in Discrete Maths by Rosen, when I came across the following question: What Geometric proposition ( having an invariant property ) does this ...
2
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2answers
116 views

If $n^2$ is even $n$ is even

I understand that there are already several answers to how to prove this question: Prove if $n^2$ is even, then $n$ is even. Prove that if $n^2$ is even then $n$ is even I am trying to understand ...
2
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1answer
50 views

Next step to reach the contradiction?

This is a problem from Discrete Mathematics and its Applications Here are my notes and my current work so far for this problem. I started with an assumption that what i am trying to prove is ...
2
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2answers
52 views

Discrete Math Proof: $A \cup B$

I'm preparing ahead for a Discrete Math course coming up this year by doing some practice problems supplemented by online notes. The problem I'm having trouble proving is the following: $A \cup B ...
0
votes
2answers
57 views

What to use for r in proof by contradiction?

This is a problem from Discrete Mathematics and its applications To this proof, I am trying to use proof by contradiction. Here is how the book described the process of proof by contradiction. I ...
4
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2answers
67 views

How to prove $( \sum_{n=1}^{\infty} |x_n|^2)^{1/2} \le \sum_{n=1}^{\infty} |x_n|$ (cauchy-product)

I am having this: $ (\ x_n)\ _{n \in \mathbb N} $ is sequence in $\mathbb C$, so the series $\sum_{n=1}^{\infty} |x_n|$ converges. I've already proved that the series $\sum_{n=1}^{\infty} |x_n|^2$ ...
0
votes
1answer
48 views

Next step to take in direct proof?

This is a problem from Discrete Mathematics and its Applications. I understand the basic ideas of the direct proof. Basically a proof is a conclusion from a series of steps to establish the truth of ...
2
votes
1answer
48 views

Next step to take in direct proof or a workaround around current dilemma?

This is a problem from Discrete Math and Its Applications I used a direct proof to do this proof. I understand the process/idea behind the direct proof, mainly (from ...
2
votes
1answer
64 views

$\exists x (P(x) \to \forall y P(y))$ [duplicate]

Prove $\exists x (P(x) \to \forall y P(y))$. Let x = y. Suppose P(x) is true. Let y be arbitrary. Since P(x) is true, it must be that P(y) is true. Since y was arbitrary, we can conclude that ...
3
votes
1answer
85 views

How to prove that $2^x,3^x,5^x\in\mathbb N$ implies $x\in\mathbb N$? [duplicate]

Let $x\in\mathbb R$ and suppose that $2^x,3^x$ and $5^x$ are all integers. Does it imply that $x$ is also necessarily an integer? I read somewhere that the answer is "Yes" and a proof is known, but I ...
0
votes
1answer
27 views

Cases for x in $ \forall x \in \mathbb{R} \exists y \in \mathbb{R} (xy^2 \neq y - x) $.

This is from Velleman p145, problem 28. Theorem: $\forall x \in \mathbb{R} \exists y \in \mathbb{R} (xy^2 \neq y - x)$. Author's Proof: Let x be an arbitrary real number. Case 1. $x = 0$. Let $y ...
1
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0answers
30 views

Logarithmic Series [duplicate]

I was doing a bit of math when I came across logarithmic series. I have no idea from where they come from. They seem so unrelated, that I have no intuition behind them at all. So, can anyone prove ...
1
vote
2answers
64 views

Evaluating $\sum_{k=0}^n \frac{1}{(2k+1)!(2(n-k))!}$

Evidently: $$\sum_{k=0}^n \frac{1}{(2k+1)!(2(n-k))!} = \frac{4^n}{(2n+1)!}$$ (says wolfram alpha) But what is a good way to come up with this?
2
votes
7answers
126 views

Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$

Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z.$ I know this is true because any even number that is squared will be even, is it also true than any even number ...
3
votes
1answer
42 views

How to derive this simple geometric relationship using cosine law?

Given the above figure, I need to show that $$cos(a_2) = \frac{x_1^2 + y_1^2 - L_1^2 - L_2^2}{2L_1L_2} $$ Where $L_1, L_2$ are the length of the red lines respectively, and $a_1, a_2$ are the ...
2
votes
2answers
97 views

Must proofs always be cited (Thesis)?

I have some proofs of theorems in my thesis that are very similar to the proofs from the literature ( "my" proofs are more extended and have more explaination, the structure isn't the same either). ...
1
vote
2answers
99 views

Proof of $\exists x(P(x) \Rightarrow \forall y P(y))$

Exercise 31 of chapter 3.5 in How To Prove It by Velleman is proving this statement: $\exists x(P(x) \Rightarrow \forall y P(y))$. (Note: The proof shouldn't be formal, but in the "usual" ...
3
votes
1answer
91 views

Convergence in dual of Sobolev space

Hi please view the following question: Consider Sobolev space $W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^{n}$ is bounded. We also have a mapping $a: \Omega \times \mathbb{R} \times ...
0
votes
2answers
26 views

The minimal polynomial divides the characteristic polynomial

How do I show that the minimal polynomial divides the characteristic polynomial? I believe I need to use the Cayley-Hamilton theorem which I understand to be The characteristic polynomial of a linear ...
1
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1answer
74 views

Prove that a function is continuous at $x = x_{0}$ using the $\delta - \epsilon$ definition

Prove that $f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$ is continuous at $0$ $\forall \epsilon > 0$, $\exists \delta = ?$ ...
0
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1answer
64 views

Prove that function $f$ is continuous at $x = x_{0}$

In class we're given the following definition about continuity, and I want to apply this definition to the problems that follow: $f$ is continuous at $x_{0} \in \mathrm{dom}(f)$ if $\forall x_{n} \in ...
2
votes
1answer
74 views

Prove that $f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$ is continuous at $0$

Prove that $$f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$$ is continuous at $0$ and discontinuous everywhere else Proof: ...
1
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2answers
66 views

prove $f$ is a constant [duplicate]

Lets's say we have a differentiable function $f:[a,b]\to \mathbb{R}$ with $f^\prime\equiv0$ How do I show that $f\equiv C$ by using the mean value theorem?
3
votes
2answers
142 views

Connectedness arguments in elementary mathematics?

To begin, let me explain a proof strategy (which I'll call the connectedness principle for want of a better, more canonical term): One way to prove that a mathematical object $O_1$ has some property ...
1
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2answers
54 views

Prove that $f(r) = 0$ for $\forall r \in \mathbb{Q} \Rightarrow f(x) = 0$ for $\forall x \in \mathbb{R}$ given $f$ is continuous

Prove that $f(r) = 0$ for $\forall r \in \mathbb{Q} \Rightarrow f(x) = 0$ for $\forall x \in \mathbb{R}$ given $f$ is continuous. I know that the Dirichlet function is discontinuous everywhere ...
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votes
1answer
55 views

Can I tell whether this expression is positive?

Can I say that the following is greater than zero $$ \frac{2 \sqrt{xy} - y}{2(z+1)},$$ when $x \leq 2y \leq z$? What if, instead, $2y \leq z \leq x $? For the second condition i.e, $2y \leq z \leq ...
20
votes
5answers
1k views

Prove that function is constant

Prove that a function $f:\mathbb{R}\to\mathbb{R}$ which satisfies $$f\left({\frac{x+y}3}\right)=\frac{f(x)+f(y)}2$$ is a constant function. This is my solution: constant function have derivative $0$ ...
0
votes
1answer
24 views

An upper bound to this fraction

The following is an expression I am trying to upper bound by a constant $$I=\frac{x}{1+2y}\leq \ ?$$ The condition that I am using is $$ 2 x < y $$ I have tried the following $$ I = ...
1
vote
1answer
39 views

How do to derive the following SIMPLE geometric relationship between two points on a plane

Can someone show why: $$x' = L_1 \cos(a_1) + L_2\cos(a_1+a_2)$$ $$y' = L_1 \sin(a_1) + L_2\sin(a_1+a_2)$$ where $L_1$ and $L_2$ are the length of the red lines
5
votes
5answers
197 views

Showing uniqueness of integers in base 3

I have recently begun self-studying Number Theory, and am working on proving: Show that every integer $n>0$ can be uniquely written as $$n = \sum_{i=0}^mc_i3^i$$ where $c_i \in \{ -1,0,1\}$ and ...
0
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0answers
45 views

Showing an equality with sequences and sets

Let $a,b$ be two real numbers and $(a_n) \subset \mathbb{Q}$ be a decreasing sequence of rationals such that $a_n \to a $. Also, take a strictly increasing sequence $(b_n) \subset \mathbb{Q} $ such ...
2
votes
2answers
64 views

How to identify an error in a proof?

Right now I'm studying how to find errors in proofs by looking for common mistakes such as circular reasoning, using examples etc. I haven't had too many problems for the most part but I've run into a ...
3
votes
1answer
67 views

Density of a set around $0$ and on $\mathbb{R}$

In this question, we prove that $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ is dense in $\mathbb{R}$ by proving that is it dense around $0$. Why is that enough to prove that it is dense on $\mathbb{R}$ ?