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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3
votes
3answers
63 views

Is it necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? [closed]

As the title suggests, is it a necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? Like, I understand the dictionary definitions of necessary and ...
1
vote
0answers
31 views

A set $A\in \Bbb{R}^n$ is compact $\iff$ every continuous function on $A$ is bounded. [duplicate]

A set $A\in \Bbb{R}^n$ is compact $\iff$ every continuous function on $A$ is bounded. I have a problem proving the direction according to which $A$ is compact. First direction I said: If $A$ is ...
1
vote
3answers
67 views

Implies in a truth table, unclear. [duplicate]

In my textbook, we have the following truth table: $P$ true and $Q$ true means that "$P \implies Q$" is true. $P$ true and $Q$ false means that "$P \implies Q$" is false. $P$ false and $Q$ true ...
1
vote
1answer
18 views

Showing an outerplaner graph has less than $2n-3$ edges

An outerplanar graph is a connected plane graph that can be drawn in such a way that all it's vertices are on the outer face. I want to show that for every $G$ outerplaner graph with $n$ vertices and ...
1
vote
5answers
147 views

If $a+b+c+d=1$ then why is the maximum value of $(a+1)(b+1)(c+1)(d+1)$ is ${\left(\frac{5}{4}\right)}^4$?

What I know is that for equations of type $x+y=8$, $xy$ attains its maximum value when $x=y$ and this can be proved by either solving the quadratic equation with completing the squares or finding the ...
0
votes
0answers
24 views

Local martingale and integral condition

Suppose $M^i_t = X^i_t - X^i_0 - \int_0^t b_i(s,X)\, ds$ where $b_i:[0,\infty)\times \Omega \to \mathbb{R}$ is a progressively measurable functional and $X^i_t: C[0,\infty)^d \to \mathbb{R}$ ( ...
1
vote
0answers
41 views

Proof of Supporting Hyperplane Theorem from basic definitions.

My purposes in posting this question are twofold. First, I would like to have a lemma which I have proven on the way to proving the Supporting Hyperplane Theorem checked for rigor (zero tolerance for ...
-1
votes
2answers
73 views

Proof that for all symmetric matrices $A$ and $B$, $AB=(BA)^T$.

Recall that a matrix, $M$, is said to be symmetric if and only if $M=M^T$. I've been trying to use the homomorphic nature of the transpose operator to prove this proposition but this approach hasn't ...
1
vote
1answer
37 views

If a set $E\subset \Bbb{R}^n$ is closed and open, then it is either $\Bbb{R}^n$ or $\emptyset$.

If a set $E\subset \Bbb{R}^n$ is closed and open, then it is either $\Bbb{R}^n$ or $\emptyset$. I have an attempt. I know, or at least think, that it is correct ideally, but I don't know how to make ...
1
vote
1answer
22 views

Confusion regarding differences between strong induction and simple induction

I don't know how to prove that any proof by induction is also proof by strong induction nor any proof by strong induction can be converted into a proof by simple induction? An example would be useful ...
4
votes
2answers
117 views

Function that is continuous and its differential is continuous

Let $ f: \mathbb{R} \rightarrow \mathbb{R}$ . Show that $f$ is continuously differentiable if and only if, for every $x \in \mathbb{R}$ there exists a $l \in \mathbb{R}$ with the property that ...
6
votes
2answers
49 views

Convergence and divergence depending on whether $n$ is odd or even

It is part of one problem I am working on: I want to prove the following conjecture ($x\ne q\pi$ where $q\in\Bbb Q$) $$\sum_{n=1}^{\infty}\frac{\sin^{2k-1}(nx)}{n^{\alpha}}\quad\text{converges ...
1
vote
2answers
63 views

Proving that $\hat{x} = x$ in least square

I am trying to show that if $Ax=b$ has a unique solution $x$, then the least square solution $\hat{x}$ is the exact one (i.e., $\hat{x} = x$). My attempt: We know that for $A x = b$ to have an exact ...
1
vote
2answers
35 views

Prove that $H=V$ if $H$ is an $n$-dimensional subspace of an $n$-dimensional vector space $V$.

Prove that $H=V$ if $H$ is an $n$-dimensional subspace of an $n$-dimensional vector space $V$. I am not exactly sure what to do to show that $H=V$. So far I have reasoned that since $H$ and $V$ ...
0
votes
1answer
20 views

How do we prove a method is optimal?

This is a very simple question, infact it's so simple that I have no idea how to solve it. We have an ordered list of $n$ words. The length of the $i$'th word is $W_i$. Our goal is to write all the ...
3
votes
1answer
31 views

$\varepsilon$ - closeness property

I'm studying Analysis from Terence Tao's book 'Analysis 1' and in an exercise he asks to prove seven properties regarding the notion of '$\varepsilon$ - closeness', which is defined as follows: ...
1
vote
3answers
164 views

Prove that there are infinity many tautologies.

For this question I think I am suppose to use proof by contradiction, but I need some hints on how to proceed with the proof. Always if someone can give me a brief explanation on how proof by ...
0
votes
0answers
11 views

Theorem implication/equivalence transitiveness in demonstrations

Suppose having three theorems $A, B, C$ that it's necessary to show being equivalent and having the following hypothesis: We know that $A \Leftrightarrow B$ and $B \Leftrightarrow C$. It would ...
-3
votes
1answer
41 views

Is this mathematical statement? [closed]

$\{\text{integers $n$ such that $n$ is even}\}$ It can be true/false so does that mean it's proposition/mathematical statement?
2
votes
4answers
227 views

How do I know which of these are mathematical statements?

While reading this book called "How to Read and do Proofs" by Daniel Solow(Google) I found the following exercise at the end of the first chapter. So how do I know if something is a mathematical ...
0
votes
1answer
80 views

Can this expression be made true ? 2 _ _ _ _ = 2015

Make this expression true: 2 _ _ _ _= 2015 The underscores must be replaced by any 2 of of the operational symbols +, - , x, / (divide). And any 2 of the digits 0,1,2..9. So, you basically need 2 ...
4
votes
2answers
140 views

Proving linearity of a function in two steps

When a function is additive, $$\color{blue}{f(x+y)=f(x)+f(y)},$$ you can extend the property to the product by an integer $$f(nx)=f(x+x+\cdots x)=f(x)+f(x)+\cdots f(x)=nf(x),$$ then to the product ...
3
votes
1answer
72 views

Another Evaluation of the Ramsey number $\mathcal{R}(3,3,3)$

The problem Show that $\mathcal{R}(3,3,3)=17$ The story behind the problem and some notation It was first proven by Greenwood and Gleason in 1955 in their paper Combinatorial relations and ...
2
votes
3answers
40 views

Elementary set theory notation verification

Reading Velleman's "How To Prove It" I came across the following expression: $$ x \in\bigcup\{\mathscr P(A)\mid A\in \mathcal F\} $$ such that $\mathcal F$ is a family of sets, $A$ is a set, and ...
2
votes
1answer
27 views

Show that $dim(X,\succeq)\leq |X^2|$ when $X$ is finite

I am trying to prove that when $(X,\succeq)$ is a finite preorder, the $dim(X,\succeq)\leq |X^2|$. Here's the full context (Exercise 11 (a)): My idea of resolution was to show that any set of ...
0
votes
1answer
27 views

Strictly monotonic increasing function with a closed domain and range

Let $a,b,c,d \in \mathbb{R}$ with $a<b$, $I = [a,b]$. Let $f: I \rightarrow \mathbb{R}$ be a monotonic, strictly increasing function. Also $c<d$ and $f([a,b]) =[c,d]$ a) Proof that $f$ is ...
1
vote
2answers
87 views

if integral $f(x)\cdot g(x)=0$ mean that $f(x)=0$?

The question: If $f(x)$ is a continuous function, such that for every continuous function $g(x)$ defined over $[a,b]$ $$\int_a^b f(x)\cdot g(x)\,dx =0$$ does it mean that $f\equiv 0$? The ...
0
votes
1answer
32 views

The image of the inverse of a continuous function

First of all I'm not sure if my title is correct with the question, I find it hard to really get about what kind of set this question is about. It would be very helpful if someone could explain this ...
0
votes
3answers
32 views

Proof:Taylor expansion of inverse trigonometric functions

I find it quite difficult to remember the Taylor expansion of inverse trigonometric functions.Actually in school we have been just taught the series (for finding limits in calculus without teaching us ...
7
votes
2answers
57 views

Basic absolute value property

Hello all I am wondering if anyone has the correct proof that I should use for Spivak calculus ( chapter 1, question 12 ) that says $$|xy|=|x| \cdot |y|$$ from past times I know it is true , but I ...
0
votes
0answers
87 views

Proof, if $f(x) = g(x)$ then $\int f(x)=\int g(x)$.

Let $f(x)$ an $g(x)$ be integrable functions over $[a,b]$ and let $\alpha$ be a point of $[a,b]$ if $f(x) = g(x)$ for all $x\neq \alpha$, then $$\int_{[a,b]}f(x)dx=\int_{[a,b]}g(x)dx.$$ So is it okay ...
0
votes
0answers
42 views

Comparison of books that teach proof techniques

I have to take discrete math and want to learn proof techniques both to get ahead in it as well as open up the possibility of understanding higher math. I've seen several books recommended such as How ...
1
vote
2answers
34 views

GCD Proof Questions

I am preparing for a discrete mathematics exam and am having trouble producing proofs for the following: 1.) Prove that $\gcd (a,c)=1 \Rightarrow \gcd (a,b)= \gcd (a,bc)$ 2.) Prove that $\gcd ...
1
vote
1answer
62 views

Prove that $\sin{\frac{2\pi x}{x^2+x+1}}=\frac{1}{2}$ has no rational roots.

Show that the following equation has no rational roots. $$\sin{\frac{2\pi x}{x^2+x+1}}=\frac{1}{2}$$ This is what I've tried: $$\left ( \frac{2\pi x}{x^2+x+1}=\frac{\pi}{6}+2k\pi ...
2
votes
3answers
34 views

Prove correctness of simple greedy algorithm to find max

We have $2n$ values $x_1,x_2,x_3,\ldots,x_n$ and $y_1,y_2,y_3,\ldots,y_n$ such that the pair $(x_i,y_i)$ represents the location of a city $i$. Assume there is no straight line that goes through all ...
1
vote
3answers
36 views

How to prove that the cross product of a countable and uncountable set is uncountable?

so my question is, how can you prove that ${\Bbb Z}$ x ${\Bbb R}$ is uncountable? So far I have tried proving that there is an uncountable subset of ${\Bbb Z}$ x ${\Bbb R}$ without luck and I'm ...
0
votes
1answer
25 views

Proof If a tree is not trivial, then there are at least two pendant vertices?

I have the following Proof but could not understand it Proof. If a tree has $n(≥ 2)$ vertices, then the sum of the degrees is $2(n − 1)$. If every vertex has a $degree ≥ 2$, then the sum will be $≥ ...
3
votes
0answers
61 views

Proving injectivity of a multivariable function

Let I denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$$ ...
4
votes
1answer
38 views

Divide a square into different parts

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with geometry, which perhaps yields the shortest, simplest proofs, but other ...
1
vote
1answer
23 views

Exlamation about a claim of an existing such cycle in a simple Graph

Suppose the following situation: this is found at (Let G be a graph of minimum degree k > 1. Show that G has a cycle of length at least k+1) Let $P=v_0v_1 \dots v_l$ be a longest path in $G$. ...
0
votes
1answer
33 views

Upper semi-continuity proof for topological spaces

Hi does anyone have any idea or a possible hint for a proof of the following result: Consider asymmetric norm $p$ on $\mathbb{R}$ given by $p(t) = t^{+}$, for $t \in \mathbb{R}$. Show that if $(X, ...
4
votes
3answers
161 views

A lot of confusion in the “Polynomial Remainder Theorem”?

Lately I've been reading about Polynomial Remainder Theorem from various sources, mainly from the wikipedea article, this post and some high school books. Wikipedea says that if we divide a polynomial ...
2
votes
2answers
81 views

Calculate probabilty that solution of the equation $x^2+px+q=0$ is in $[0,5]$ [closed]

Calculate the probabilty that the solution of the equation $$x^2+px+q=0$$ is in the interval $[0,5]$.
1
vote
1answer
62 views

If $G$ is simple and $deg^+(v)\geq k \geq 1 \space \forall \space v \in V$ there is a simple cycle of at least size $k+1$

I have the following proof but it is tough could someone help me to understand it, Proof: Start at an arbitrary node $v$ and mark it, and so on until you have marked all nodes in the series then a ...
0
votes
2answers
53 views

Proof that if all vertices have degree at least two then G contains a cycle

Here is the proof, but please correct me if wrong : We assume $G$ is simple and let $P$ be the longest path $=v_0v_1v_2\ldots v_{a-1}v_a$. As it is given that the degree of $v_a$ is even ,then $v_a$ ...
0
votes
1answer
32 views

Riemann-integral of a non-continuous function

Let $f : \,\mathbb R \to \mathbb R$ be a function with a discontinuity at point $x_0$. How can I prove formally that $f$ the Riemann-integral of $f$ exists, i.e. that $f$ fulfills $\sum_k (\sup ...
3
votes
5answers
125 views

Simple question with a paradox

"I have three boxes, each with two compartments. One has two gold bars One has two silver bars One has one gold bar and one silver bar" You choose a box at random, then ...
0
votes
1answer
56 views

What is the wrong in proving this Assumption?

In the famous case of proving that total number of degrees in a graph $G$: $\sum \deg(v_G) =2m$. By Using Proof by induction:- for: $$m=0: 2m= 2*0 =0 \tag 1$$ is true .. $(2)$...We add a new edge to ...
0
votes
3answers
77 views

Show that: $\int_{0}^{\infty}{x^2e^{-x^2}}{dx} = \frac{1}{2}\int_{0}^{\infty}{e^{-x^2}}{dx}$

I am fully uncertain of how to approach this problem: Show that: $$\int_{0}^{\infty}{x^2e^{-x^2}}{dx} = \frac{1}{2}\int_{0}^{\infty}{e^{-x^2}}{dx}$$ We've just completed the section on improper ...
1
vote
1answer
56 views

Methods to prove that a function is continuous

Although I seem to understand the concept of continuity in connection with functions, I am often stuck proving that particular functions are continuous. I think the epsilon-delta definition is the ...