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
votes
2answers
51 views

How do I show :$\sum_{n=0}^{\infty }\frac{(-1)^n}{n+1}=\ln2$? [closed]

How do i show this : $$\sum_{n=0}^{\infty }\frac{(-1)^n}{n+1}=\ln2\text{ ?}$$ Thank you for any help
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
34 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
21 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
46 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
35 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
20 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 $G$. ...
-1
votes
0answers
32 views

Starting to proof… Fundamental Theorem of Arithmetic [duplicate]

I have the following: $$ m^2 = 3 n^2 \quad\text{where } m, n \in\mathbb{Z} \text{ and } n \neq 0 $$ I know how the Fundamental Theorem of Arithmetic works, every number is a unique product of ...
0
votes
1answer
28 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, ...
3
votes
3answers
122 views
+50

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
80 views

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

Calculate the probabilty that the solution of the equation $$x^2+px+q=0$$ is in the interval $[0,5]$.
1
vote
1answer
58 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
48 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
29 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
120 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
55 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
1answer
31 views

what are all types of Graphs ? [closed]

As I know there are many types of graphs: 1- Simple Graph 2- Completed Graph 3- Directed Graph 4- UnDirected Graph 5- Strongly Connected Graph 6- Weakly Connected Graph 7- Euler Graph 8- Cubic ...
0
votes
3answers
76 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
53 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 ...
8
votes
3answers
329 views

Need Suggestions for beginner who is in transition period from computational calculus to rigorous proofy Analysis

I have completed basic calculus 1,2,3 courses, Linear Algebra, etc. I have not, however, got into rigorous Analysis yet, which I am planning to do now. I have three books in mind. They are : Terence ...
0
votes
0answers
17 views

Functions with 0 as an interior point of the domain

We have $f,g,h : \mathbb{R} \rightarrow \mathbb{R}$, with $0$ an interior point of the domain. Let $k \in \mathbb{N}$. We use the following notation $h = \mathbb{O}(x^k)$ if the open interval $ I ...
1
vote
1answer
21 views

Intermediate theorem on a function of more dimensions

I have the following question in my textbook: Let $M>0$ and $\phi : \mathbb{R}^p \rightarrow \mathbb{R}$ be differentiable with the property that $$||\nabla \phi(x)|| \leq M$$ Proof that ...
0
votes
1answer
38 views

Riemann integral of a non continuous function

We have a function $f : I=[0,1] \rightarrow \mathbb{R}$ defined as: $$f(x)=\begin{cases} 1 &\text{if }x\in \mathbb{Q} \\ 0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}$$ a) Show ...
0
votes
1answer
63 views

Riemann Integral

I tried to do the following excercise Let $a,b \in \mathbb{R}, a<b$. We have a bounded function $ f: [a,b] \rightarrow \mathbb{R}$ which has an integral or in other words, there exists a ...
0
votes
1answer
34 views

If $G$ is simple and $deg_+(v) \ge k\ge 1$ , then there is a simple cycle of at least size $k+1$

I am going to show you my proof/ and please correct me if wrong: Begin with some node $v$, and mark it. Follow one of its outgoing edge $(v,w)$ to next unmarked node, and mark it, by doing this ...
0
votes
1answer
38 views

Strong Induction: Prove that sqrt(2) is irrational

This question comes directly out of Rosen's Discrete Mathematics and It's Applications pertaining to Strong Induction. Use strong induction to prove that $\sqrt{2}$ is irrational. [Hint: Let $P(n)$ ...
3
votes
1answer
39 views

Arrange 1-12 around a circle

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 plain old algebra, which yields the shortest, simplest proofs, but other ...
-3
votes
1answer
24 views

Proof related to $2 \times 2$ matrix [closed]

Let $A$ be a $2 \times 2$ matrix such that $AX = XA$ for all $2 \times 2$ real matrices $X$. Show that $A =kI$ for some $k$ belonging to $\mathbb{R}$
2
votes
2answers
41 views

Proofs utilizing the Well-Ordering Property

This question comes directly from as an example in Chapter 5.2 of Rosen's Discrete Mathematics and It's Applications textbook on page 341. Use the well-ordering property to prove the division ...
-2
votes
2answers
36 views

Proof related to matrix [duplicate]

Let $A$ and $B$ be $n \times n$ real matrices such that $A^2 = I, B^2 = I$ and $(AB)^2 = I$. Prove that $AB = BA$. Someone help me with this problem
0
votes
1answer
32 views

Proof related to matrix with if and only if condition

Suppose $A$ and $B$ are matrices such that $AB$ and $BA$ are defined. Prove that $$(A+B)^2=A^2+B^2+2AB\quad\text{if and only if}\quad AB=BA.$$ Someone help me with this.
2
votes
5answers
75 views

Show that if $x \ge 1$, then $x+\frac{1}{x}\ge 2$ [duplicate]

So here the problem goes: Show that if $x \ge 1$, then $x+\frac{1}{x}\ge 2$. This is a very interesting word problem that I came across in an old textbook of mine. So I know it's got something ...
0
votes
0answers
16 views

Proofs By Induction Help

Hey I'm having some problems on these proofs. I think Im doing right but if anyone can show me the right way to do them that would be great! 1) ∑ i=1, n of (2i) = n^2 + n ...
11
votes
1answer
205 views

How to combine the four Theorems in order to prove the statement?

I have a question concerning a statement about Random Walks on $\mathbb{Z}$. Let $F$ be a distribution on $\mathbb{Z}$ which has mean $0$ and finite variance. Let $\left\{X_1,X_2,\ldots\right\}$ be an ...
2
votes
2answers
34 views

Proving binary integers

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 binary integers (For ${0, 1, 2, 3}$ we have the representations $0, 1, 10, ...
14
votes
8answers
1k views

Variation on Pythagoras: If $a^2 + b^2 = c^2$, then $a + b \leq c\sqrt{2}$

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 derivative of the Pythagorean Theorem using calculus, trigonometry, ...
0
votes
1answer
82 views

Proving that $\sum \deg(v) = 2m$ for any Graph $G$

Here is My proof, please correct me if wrong, I try to be formal. Proof by Induction: Let $\sum \deg(v)=2m$ assumption... when #of nodes is $n=0$. so here the equation is $\sum \deg(v)=2(0)=0$ ...
0
votes
2answers
74 views

Explain how the proof is done

A solution of matrix problem appears to be as follows some one explain the following in the solution why is A cube is eliminated and fourth power of A is obtained? In the seventh line In the ...
1
vote
6answers
121 views

How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$

How to prove $$1+x \leq e^x~\forall x \in \mathbb{R}$$ I'm stuck, I tried taking logs but didn't know how to proceed.
6
votes
2answers
110 views

What does “rigorous proof” mean?

I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more ...
0
votes
1answer
40 views

Help with a (simple?) proof

I have the following problem: I want to prove under what conditions the value of a cell (e.g. 1 or a in the graphics) is the column mean (e.g. 1.5) plus row mean (2) minus the grand mean (2.5) like ...
0
votes
1answer
42 views

Prove that the hypotenuse is the longest side in a right triangle. How to write a formal proof for something so obvious?

From trig text. Given hint: is $ a^2 + b^2 > a^2 $. Pythagorean theorem,obviously. What would be an acceptable proof?
1
vote
1answer
35 views

Maximal Distance of a graph

So I have the following graph $G = (V, E)$ with $V = [d]^n$ and $E = \{\{(a_1,\dots , a_n), (b_1, \dots , b_n)\} | a_2 = b_1 a_3 = b_2, \dots a_n = b_{n-1}\}$. That is called a $(n, d)$-dimensional ...
1
vote
5answers
88 views

Prove that $ a^2-4b \neq2$ if $ a,b \in \mathbb{ Z}$

My solution : We suppose that is true. Then by contradiction: $a^2-4b-2=0$ $a^2=4b+2$ $a=2(b+1/2) ^{0.5}$ then $(b+1/2)$ is fraction and rooted by $0.5$ so the square root of any fraction $+$ ...
-4
votes
1answer
112 views

I have proved that 1 + 1 = 0 [closed]

I have proved that 1 + 1 = 0 in one of my questions where a field was given. I was wondering if it is true in every field we have 1 + 1 = 0. Also i was wondering (i know how to prove 1 + 1 = 0) can ...
-1
votes
2answers
47 views

Every field has at least two elements

I got a question saying in every field (F, +, ⋅, 0, 1), the set F has at least 2 elements. It asks if it is true prove it or if false provide a counterexample. I understand the idea of finite fields ...
0
votes
2answers
42 views

Field Question Proof with Axiom 4

Prove that if $(F,+,⋅,0,1)$ is a field, then there is no element $w ∈ F$ such that $0 \cdot w = 1$. Note that Axiom 4 from lecture (aka "M4" in the textbook) ensures that for $x ≠ 0$, there is a $w ∈ ...
1
vote
1answer
53 views

Proof of definition of invertible matrices [closed]

Let $A \in \Bbb R^{n \times n}$. Then $A$ is invertable if and only if a Matrix $B \in \Bbb R^{n \times n}$ exist such that $AB=E$. This seems like the definition of an invertible matrix but how ...
0
votes
0answers
30 views

Determining if two bounds are true

Question says assume $f$ and $g$ have a domain of the integers, and target space of the real numbers. $f$ and $g$ are bounded. Prove if the following statements are true or give a counterexample: if ...
2
votes
1answer
130 views

Strategy/Proof behind the Perfect solution of a Multiplication Game

So the below is the question Question: Jacob and Vicky play the fun game of multiplication by multiplying an integer p by one of the numbers 2 to 9. Jacob always starts with p = 1, does his ...