2
votes
2answers
34 views

Proving corollary to Euler's formula by induction

I'm currently looking at two proofs to the following corollary to Euler's formula and I'm not quite seeing how the authors can make a specific assumption in their proof. One proof comes from my ...
1
vote
1answer
33 views

Prove that if graph $G$ is a 3-connected planar graph then its dual must be simple.

I'm trying to study for a quiz. I think I'm on the right track with this problem, however, I'm having a difficult time formalizing it. Prove that if graph $G$ is a 3-connected planar graph then its ...
0
votes
2answers
40 views

Show that $A^{(x,y)}$ is countable.

Question: Let $A$ be a countable set $A^{(x,y)}$ the set of all functions from $(x,y)$ to $A$. Show that $A^{(x,y)}$ is countable. My attempt: By proposition 7.1.2iii, $\mid B \mid^{\mid A \mid}$ ...
1
vote
1answer
27 views

The Open Set $X-\lbrace x \rbrace$

I am task with proving the following: if $x \in X$ then $X- \lbrace x \rbrace $ is an open set I kind of have an idea but I am unsure about it and how to express it. I was thinking about using the ...
0
votes
0answers
12 views

geometry 2 column proof of tangent chord angle corollary

I need to prove 12.23 on this section (http://i.imgur.com/M5iev9K.png) I can use any of the theorems or corollaries before 12.23 but not the ones after it. This is a list ...
5
votes
0answers
35 views

My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
2
votes
1answer
110 views

Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
1
vote
0answers
39 views

My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.

What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Among three elements ...
3
votes
0answers
57 views

My first simple direct proof (very simple theorem on real numbers). Please mark/grade.

What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Let $I = [a,b]$ be a non-empty closed ...
2
votes
0answers
38 views

My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade.

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
17
votes
3answers
663 views

My first induction proof (very simple number theory). Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
3
votes
1answer
47 views

Functional form of a solution to a Differential Equation (Euler-Lagrange)

Let $f=f(q(t),\dot q(t),t)$, where $q(t)=\{q_1(t),...,q_N(t) \}=\{q_{a}\}_{a=1}^N$ and $\dot q:=\frac{dq}{dt}$. I want to show that if the following equations (Euler-Lagrange) are satisfied ...
1
vote
1answer
50 views

Prove thoroughly: If the degree of all vertices is greater or equal to $\frac{|V| - 1}{2}$, then the simple graph is connected.

I am struggling to write a good, thorough proof. The proof is supposed to be logically rigorous, correct and complete (e.g. no hidden assumption). Moreover, style is important - the proof should be ...
2
votes
1answer
49 views

Metric Spaces: The dist function

Given that $A$ is defined as non-empty subset of $(X,d)$ The distance function is defined as such: $dist(x,A)=$ inf $_{y\in A} \lbrace d(x,y) \rbrace $ Given the above we are asked to prove the ...
1
vote
2answers
33 views

Show the equivalence: $ab|c \iff a|c$ and $b|c$

Let $a,b \in \mathbb{Z}$ \ {$0$} with $gcd(a, b) = 1$ and let $c \in \mathbb{Z}$. Show the equivalence: $ab|c \iff a|c$ and $b|c$ Also give an example of numbers $a,b \in \mathbb{Z}$ \ {$0$} and ...
1
vote
1answer
62 views

Possible book correction or am I missing something?

Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
20
votes
5answers
3k views

Is a brute force method considered a proof?

Say we have some finite set, and some theory about a set, say "All elements of the finite set $X$ satisfy condition $Y$". If we let a computer check every single member of $X$ and conclude that the ...
1
vote
2answers
61 views

$\forall x \in \mathbb{R}$ show that $x=\sum_{n=1}^\infty k_na_n = \prod_{n=1}^{\infty}m_na_n$ …

Yet again, another cool problem from the book "problems in mathematical analysis" by Piotr & Witkowski: Prove that if $a_n \neq 0$, $n=1,2,\cdots$ and $\displaystyle \lim_{n \to \infty} a_n = 0$, ...
4
votes
1answer
39 views

Problems with a proof that -in a linear order- a minimal element is the smallest element

I have a problem with a proof I found in Velleman's "How to prove it". This is sort of interesting, because it is the very first time I cannot see the structure of a proof presented in the book. The ...
1
vote
3answers
54 views

Let $P(x)$ be any polynomial and suppose that $a_n \rightarrow a$. Prove $lim_{n\rightarrow\infty} P(a_n) = P(a)$

I know the limit rules but that's not helping me out much here this seems so simple but I don't even know where to start. I read the chapter on this and they don't do any examples like this. I ...
0
votes
6answers
92 views

Suppose that $x$ and $y$ satisfy $\frac{x}{2} + \frac{y}{3} = 1$. Prove that $x^2 + y^2 > 1$.

Ok , i tried to prove this via Contrapositive setting $x^2 + y^2 \le 1$. After doing some algebra i have arrived at $x \le \sqrt{-y^2}$. I'm fairly sure this isn't right. I also solved for x and y in ...
0
votes
1answer
47 views

Struggling to prove that if $n$ is a non zero integer, and $m > 0 \mid n$ then $m \leq |n|$

i need to prove that if $n$ is a non zero integer, and $m > 0$ and $m \mid n$ ($m$ divides $n$), then $m \le |n|$. I feel like i can do it by a combination of proof by contradiction and cases (ie ...
1
vote
3answers
33 views

EXCERSICE VERIFICATION: Find where $f(x):=|x|+|x+1|$ is differentiable and calculate its derivative

Could someone verify my excersice? a) $f(x):=|x|+|x+1|$ First, analyse the roots of each absolute value, where they go to zero: $$|x|:=\left\{\begin{matrix} & x& x>0 \\ & x- ...
0
votes
1answer
103 views

Comparison Theorem for Integrals

Problem: Let $a>0$ and $b>a+1$. Use the Comparison Theorem to show that the following integral is convergent: $$\int ^ \infty _0 \frac{x^a}{1+x^b} \ dx$$ My attempt at this was that since ...
0
votes
0answers
23 views

Understanding 2nd half rank-nullity theorem proof.

I'm trying to understand the second half of the rank-nullity theorem (the part that shows $T(e_{k+1}) \dots T(e_{k+r})$ is independent). Assume $e_1 ,\dots e_k, e_{k+1}, \dots e_{k+r}$,is a basis for ...
0
votes
0answers
31 views

Ideal Test Proof

Let $\emptyset \subset I \subseteq R$. Prove that I is an ideal of R if and only if $a-b, ra, ar$, $\in$ $I$ for all $a, b \in I$ and $r \in R$. I know that if I is an ideal in a ring R and $a \in ...
2
votes
1answer
50 views

Integral Domain and no nonzero divisors Proof

Prove that a commutative ring is an integral domain if and only if it has no nonzero zero divisors. I think my main problem is that I'm getting jumbled in the wording! By 'no nonzero zero divisors' ...
2
votes
1answer
35 views

Does all non-monotonic continues functions have $x_0 \in \mathbb{R}$ such that $f'(x_0)=0$?

Given $f\colon\mathbb{R} \to \mathbb{R}$, $f$ is differentiable on $\mathbb{R}$ and the $\lim_{x \to \infty}f(x)$ does not exists . show/prove formally that there exists $x_0 \in \mathbb{R}$ such ...
0
votes
1answer
58 views

Prove this limit

If $\lim_{x\to a}f(x)=L>0.$ Prove $\lim_{x\to a}\sqrt(f(x))=\sqrt(L)$. I know that we have: |$\sqrt(f(x)-\sqrt(L)|=|(f(x)-L)/\sqrt(f(x)+\sqrt(L)|\le|(f(x)-L)/L|<|(f(x)-L)|<\epsilon$. ...
0
votes
2answers
118 views

How to prove the equation |xy|=|x||y| if we assume x and y are real numbers by using analysis. [closed]

Prove that if x and y are real numbers, then |xy|=|x||y|. Hint check all the cases. I tried assuming the left hand side equals the right hand side if we remove absolute values. Also, tried using the ...
3
votes
3answers
128 views

Proof of something that doesn't exist

Let $\lfloor x \rfloor$ be the greatest integer function. Show that the $\lim_{x\to 2} \frac{1}{\lfloor x \rfloor}$ does not exist. So far I have: Assume the limit exists. Choose $\epsilon ...
0
votes
1answer
45 views

How to show that if $ p \equiv 1,3 \pmod 8$ then there exists a $u,v \in \mathbb Z: u^2 + 2v^2 = p$

I'm trying to show this statement: $$p \equiv 1,3 \pmod 8, \; \; \exists \; u,v \in \mathbb Z : u^2 + 2 v^2 = p.$$ I believed I proved it the other direction using the ring $\mathbb Z{\sqrt{-2}}$. ...
2
votes
2answers
47 views

My problem in understanding the minimal counterexample technique

If minimal counterexample method of proof is to assume to opposite of an argument is true and then finding a counterexample for the opposite and then concluding the validity of the original argument, ...
0
votes
1answer
16 views

Ideals of set of functions from real to real

I'm looking to prove the following is an ideal of the set of functions from real numbers to real: a)the set of all f such that f(x) = 0 for every rational x b) the set of all f such that f(0) = 0
0
votes
1answer
48 views

$A=(A \cap B) \cup(A \cap B^\mathsf{c}) $

I would like to know if this proof is correct. If not, what would I have to change to make it rigorous? This set equality seems really obvious, and because of that I am not sure if I have given enough ...
0
votes
1answer
33 views

Proof on Open & Closed Sets

I just did a quick proof, and it seemed so simple that I wanted to check if it was correct. Prove that if you have a nonempty subset, $S$, of a domain $\Omega$, and $S$ is both open and closed, then ...
1
vote
0answers
28 views

Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
2
votes
4answers
82 views

$f:X\to Y$, $A,B,\subseteq X$. Show that $f(A\setminus B)=f(A)\setminus f(B)$ iff $f(A\setminus B)\cap f(B) =\emptyset$

I tried to prove this but I am not sure if its correct. Please help me out with any tips or advice on how to improve. Here it is: First let $f(A\setminus B)\cap f(B)=\emptyset$. Now $$f(A\setminus ...
0
votes
2answers
60 views

Absorbing Element is a Unit

Show that an absorbing element of a monoid is a unit if and only if it is the only element. This is an if and only if proof so that means I have to prove it both ways: A implies B and B implies A. ...
1
vote
2answers
85 views

prove or give a counter-example

I think I have solved it (please check) but I would like to see and (re)-learn how one writes a proper proof (including the mathematical signs) and little things (I might have missed), maybe even more ...
0
votes
0answers
45 views

The Toad and Frog Game - Proof by Inducation

Toads and Frogs is played on a 1 × n strip of squares. At any time, each square is either empty or occupied by a single toad or frog. Although the game may start at any configuration, it is customary ...
0
votes
0answers
24 views

Equivalence formulation of continuity

Proposition (Equivalence formulation of continuity): Let $X\subset \mathbb{R}, f:X \rightarrow \mathbb{R}, x_0\in X$. Then the following statements are logically equivalents. (1) For every sequence ...
4
votes
3answers
74 views

Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction: How to prove one of them?

Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction How to prove one of them ? On Proofwiki there is an article proving the equivalence of the ...
1
vote
2answers
63 views

What is wrong with the proof given below?

This problem comes from Solow's book, 2nd edition. What is wrong with the proof given below? If $r$ is a real number with $|r| \leq 1$, then for all integers $n \geq 1, 1 + r + r^2 + \ldots + ...
2
votes
2answers
53 views

Upper bound of a set (equivalence problem)

I took a real analysis course three years ago and I unfortunately didn't get all of it, starting with basics. Question: Let $E$ be a set of real numbers. Show that $x$ is not an upper bound of $E$ ...
2
votes
0answers
60 views

Set of limit points of S is closed in a metric space X

A point $x \in X$ is a limit point of a subset S of X, if every ball $B(x;\varepsilon)$ contains infinitely many points of S. Show that x is a limit point of S iff there is a sequence {$x_{j}$} ...
6
votes
1answer
188 views

Intuition behind proof and verification partially ordered sets

Hi everyone in the book that I read I have trouble to understand the argument of the proof at the below proposition. There is a lot of point which are left as exercises, which is great. One of these ...
1
vote
1answer
67 views

Find the index of the equilibrium points of the system (Question on solution)

I have the following system: $$\dot{x} = 2xy$$ $$\dot{y} = 3x^2-y^2$$ I have the following solution: The system has one equilibrium point at the origin. Let the curve $\Gamma$ surrounding the origin ...
1
vote
1answer
120 views

prove Turing recognizable

This is actually an old exam question its not my homework; Let L = { : M is a TM with an input alphabet of {a,b} and M accepts at most one word, i.e. M either accepts no words or accepts exactly one ...
9
votes
1answer
184 views

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...