For questions about the formulation of a proof, not about the mathematics behind it.

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Is my parsing to symbolic logic of this statement correct?

Statement Prove that the natural number x is prime iff x > 1 and $\sqrt x$ there is no posi- tive integer greater than 1 and less than or equal to x that divides x. My parsing attempt into ...
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0answers
27 views

Proof by induction of Gronwall's inequality

I've an exercise which is the following: Gronwall’s Inequality Let $A > 0, B \geq 0$. Let $(\epsilon_j)_{j \in \mathbb{N}}$ be a sequence of real numbers with $$|\epsilon_{j+1}| ...
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4answers
43 views

Proof that $A + 1 \leq e^A$ for all $A > 0$

I was reading a proof where at a certain point the prover uses the following inequality $$A + 1 \leq e^A$$ which in my opinion needs also a proof to be used around. I think I'm missing some ...
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3answers
24 views

How to prove a function from A to B

I have a question that says... THEOREM: The function $f: \mathbb{R}_{\geq 0}\rightarrow \mathbb{R}$ given by $f(x) = ln(x)$ is onto. If you were going to prove this statement, what is the first ...
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2answers
36 views

How to disprove a theorem

I have a question that says, Explain how to disprove a theorem of the logical form "$\forall x \in A, P(x)$". Write the logical form of the statement you want to prove. So disprove a theorem, ...
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1answer
17 views

How to improve my proof and whether or not one condition in the statement is important in writing the proof?

A simple graph $G$ is connected iff for every partition of the vertices into two non-empty sets $X$ and $Y$, there is a vertex $x\in X$ and a vertex $y\in Y$ such that $xy$ is an edge of $G$. My ...
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0answers
31 views

How to prove that $p$ divides $a^p -a$ for every integer $a$. [on hold]

How to prove this Fermat's little theorem: $p$ divides $a^p -a$ for every integer $a$.
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Can someone tell me if constitutes enough proof to solve this infinite product?

I have a project do for my Calc II class where we must prove that $\lim_{n\to\infty}\prod_{k=1}^n(1-a_k)=0$ where $\{a_k\}_{k=1}^\infty$, $1>a_k>0$, $\sum_{k=1}^\infty a_k=\infty$. ...
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2answers
18 views

Using the definition of the convergence of the sequence, how can I prove that this sequence converges to the limit?

I suppose what I am confused most on here is the algebraic method. $\lim_{x\to\infty} $ $ 2n^2/(n^3 + 3) $ = 0 I have set it up so far: Let $ \epsilon > 0 $ be given. I set up the equation ...
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1answer
19 views

Proving well definedness of addition in real numbers. Real numbers defined as infinite decimal expansions.

As the title says. I have to prove that the addition in the real numbers is well defined. Here are the definitions of both the real numbers and the addition of real numbers. These are translations. ...
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1answer
46 views

Trying to Understand How to write Proofs

I am trying to study for a proofs final, and I'm really struggling with writing proofs. Does anyone have any suggestions that might help me to write proofs when given a theorem? I know there are ...
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3answers
42 views

What Proof Strategy to use

I have this theorem(see below) that I am trying to prove. However, I am struggling with how to get started; I don't understand what which proof strategy to use like proof by contradiction, if P then ...
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1answer
27 views

Showing $\mathbb{B}_{\mathbb{Q}}$ is a bases for $\mathbb{R}_{\text{usual}}$

Show that the collection $\mathbb{B}_{\mathbb{Q}} := \{(p, q) \subseteq \mathbb{R} : p, q \in \mathbb{Q}, p < q \}$ is a basis for the usual topology on $\mathbb{R}$. Solution: We know that ...
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2answers
27 views

Is $h(x)=\left(x^2; 1_{[0,\infty)}(x)\right)$ an injective function?

Let h defined by : $$\begin{align} h \ \colon\ \mathbb{R} & \to \mathbb{R}^{2} \\ x & \mapsto h(x)=\biggl(f(x);g(x)\biggr). \end{align}$$ and $$\begin{align} f \ \colon\ \mathbb{R} ...
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2answers
23 views

Check my proof on showing a graph with each vertex's degree at least $e$ has every tree with $e$ edges a subgraph

Let $T$ be a tree with $e$ edges and $G$ be a simple graph such that ech vertex has degree at least $e$. We need to show that $T$ is a subgraph of $G$. I tried to prove this by induction. The base ...
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0answers
39 views

Topology bases for $\mathbb{R}_{\text{usual}}$

I'm trying to compile correctly formulated solutions to common topology questions as a summer project. I'm not very confident in my proof writing abilities so I'm going to post my solutions here for ...
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3answers
21 views

Given $A \subseteq X$ in the discrete and the trivial topology, find closure of $A$

Given $A \subseteq X$ in the discrete and the trivial topology, find closure of $A$ Note the definition of closure I am using is one in Munkres: $x \in \overline A \iff \text{ for every ...
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2answers
34 views

Let $n$ be a natural number s.t. every natural number less than $n/2$ divides $n$. Prove $n$ is less than or equal to $6$.

From an intro to algebra text i'm reading. Need hint on how to go about proving it.
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2answers
85 views

How can I prove this equation holds?

As the final part of a big proof I got for uni homework: (It is an extra question, may be unsolvable) $$k^n<\sum_{i=0}^n\binom{n}ik^{n-i}(2^i-1)$$ My idea is to develop the right side into an ...
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1answer
23 views

$\mathbb{Z}\setminus U$ is open, where U is a basic open set of $\mathcal{B}$, the set of all arithmetic progressions

Let $m, b \in \mathbb Z$ with $m \neq 0$, and $U$ is of the form $Z(m, b) = \{ mx + b \mid x \in \mathbb Z \}$ I'm not sure how to show $\mathbb{Z}\setminus U$ is open, I was thinking to expressing ...
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0answers
36 views

Prof claimed that $\arcsin x + \arccos x$ does not always round to 90º under the same number of significant figures.

The core problem was simple: Determine the interior angles of triangle ABC given side $a = 12.34cm$ and hypotenuse $c = 35.32cm$. Simple. Clearly, the solution is: $\sin A = ...
3
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2answers
37 views

Could anybody please check my proof about connected graph?

I have written a proof of the following statement but not sure whether it is correct or not. Let $G$ be a connected graph and each vertex has even degree. Show that if we remove ANY edge of the graph ...
2
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1answer
39 views

How to prove $\lim_{x \to \infty} \dfrac{x^a}{x^b} = 0$ for all nonnegative constants $a < b$.

I'm working my way through Mathematics for Computer Science at MIT OCW, and there is a lemma in the text that I am trying to prove and I've gotten stuck. $$\lim_{x \to \infty} \dfrac{x^a}{x^b} = 0$$ ...
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0answers
38 views

Help complete this proof on transcendentalism

Proof $\pi*e$ is transcendental. either $\pi + e$ or $\pi*e$ is transcendental to see take $(x-\pi)(x-e)=x^2-(\pi+e)x+\pi*e$. Case 1 assume $\pi$ and $e$ are algebraically independent. It follows ...
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1answer
17 views

In a directed graph with n≥2 nodes, if two different nodes reaches every nodes (including itself), then this graph is strongly connected.

I think this statement is true because if node a can reach every node (including node b) and node b can reach every node (including node a), there is an edge between node a and node b. This means that ...
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0answers
31 views

Is the composition of uniformly distributed functions uniformly distributed?

Let $\mathcal{I}:=[0,1]$. Def: A measurable function $\varphi:\mathcal{I}\rightarrow \mathcal{I}$ is said to be uniformly distributed with respect to the Lebesgue measure $\Lambda$ if, for any ...
2
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1answer
27 views

The composition of measurable function is not measurable: only for Lebesgue-measurability?

Let $\mathcal{I}:=[0,1]$. Let $\mathcal{R}(f)$ denote the range of a function $f$. Let $\Sigma$ be the $\sigma$-algebra of $\mathcal{I}$. Consider the measurable and continuous functions ...
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1answer
35 views

How to fill in the gaps in my proof to make it more convincing?

Let $T$ be a tree with $3$ edges. Let $G$ be a simple graph such that each vertex has degree at least $3$. Show that $G$ has $T$ as a subgraph. This statement is obvious but I am not sure how to ...
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0answers
41 views

In a directed graph with $n \geq 2$ nodes, if two different nodes reaches $n$ nodes, then this graph has a directed cycle.

I think this statement is true because if first node can reaches every other node (including second node) and second node can reaches every other nodes (including first node), then first node and ...
2
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0answers
35 views

Prerequisites to understanding proof of Fubini's Theorem? [on hold]

I'm currently studying tensor analysis, and I have studied elementary calculus (meaning calc I, II, III, and diffy Q), as well as linear algebra. Given all of this, what are the rest of the required ...
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1answer
31 views

What is a useful starting idea to think about this simple graph problem?

I would like to prove the statement that there are $2^{\binom{n-1}{2}}$ simple graphs are there with vertex set $\{1,\ldots,n\}$ such that every vertex has even degree. The thing that confuses me is ...
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1answer
19 views

Writing a proof for $f(W) \setminus f(X) \subseteq f(W\setminus X)$

I am trying to write a proof to prove/disprove the following question: Will it always be true that $f(W\setminus X) = f(W)\setminus f(X)$? I know to prove this you need to show both ways since ...
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1answer
12 views

$\mathcal B_{\mathbb Q}$ = = { [p, q) ⊆ R : p, q ∈ Q, p < q } is not a bases for the Lower Limit Topology

I'm having a bit of trouble proving this: The definition of Lower Limit Topology I am working with: $ \{[a, b) \subseteq \mathbb R \ \text s.t \ a < b\}$. The only thing I can think of is that ...
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1answer
15 views

Is this proof about the null space and column space correct?

My question asks me to show that if $A$ and $B$ are $n\times n$ matrices, and $AB=0$, then the column space of $B$ must be a subspace of the nullspace of $A$. My attempt at a proof is like this: we ...
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1answer
31 views

Show that $\bigcap_{r>0}B(a,r)=\bigcap_{n=1}^\infty B(a,\frac{1}{n})=\{a\}$

Let $(M,d)$ be a metric space. Show that $\bigcap_{r>0}B(a,r)=\bigcap_{n=1}^\infty B(a,\frac{1}{n})=\{a\}$ where $B(a,r)$ is a ball with center in $a$ and radius $r$. My attempt: Set $0<r\leq ...
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3answers
71 views

There is no uncountable collection of pairwise disjoint open sets in $\mathbb R$

Working in $R_{\text usual}$ Topology: Show that there is no uncountable collection of pairwise disjoint open subsets of $\mathbb R$. Definition of $R_{\text usual}$ I'm working with: $\{U \subseteq ...
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1answer
47 views

Not sure how to prove this statement by contradiction?

There is this a simple looking and intuitive statement but I am not sure how to start approaching this problem. Let $S=\{s_1,s_2,\ldots,s_n\}$, where $s_1,s_2,\ldots,s_n>0$ such that ...
0
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1answer
40 views

How to prove the group of roots of unity in $\mathbb{C}$ is a group

I mostly need help with proving $G$ is closed but a verification of the other parts is appreciated. Let $G = \{z \in \mathbb{C} \mid z^n=1$ for some $n\in \mathbb{Z^+}\}$ I want to start by proving ...
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2answers
62 views

How to prove $f^{-1}(f(X)) = X$

Suppose $X \subseteq A$. Will it always be true that $f^{-1}(f(X)) = X$? I am try to prove this problem with either proofs or counterexamples. I have found a counterexample for $f^{-1}(f(X)) ...
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1answer
33 views

Prove (or check) the expression is positive given constraints on variables?

The following proof problem have taken me a few days. Perhaps it is too hard for me to overcome it. Can you help me? The expression is by the following: \begin{equation} \begin{split} ...
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1answer
41 views

Stuck with proofs [closed]

Can someone please show me how to write a proof for this. Theorem $5.4.2.$ Suppose $f: A \rightarrow B$, and let $W,X \subset A$. Then $f (W \cap X) \subseteq f(W) \cap f(X)$. Furthermore, if $f$ ...
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1answer
48 views

When to use $\in$ and $\subseteq$ when talking about bases and topologies

Can someone demonstrate a concrete example of when to use $\in$ and $\subseteq$ when talking about topologies and bases? When is something $\subset$ of a basis or a topology and when is something ...
0
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1answer
14 views

Under what assumptions on φ is Tco-φ a topology

Fix a set X, and let φ be a property that subsets A of X can have. Define Tco-φ = {U ⊆ X : A = ∅, or X \ U has φ } . Under what assumptions on φ is Tco-φ a topology on X? What I think: 1. X\X has φ ...
3
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1answer
41 views

Find the mistake of the following inductive proof: all algorithms have the same time complexity

I came across this problem: Find the mistake of the following inductive proof: Theorem: all algorithms have the same time complexity. Proof: (By induction on the number of algorithms.) The ...
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2answers
25 views

Prove $A\setminus(B \cup C) = (A \setminus B) \cap (A \setminus C)$ using element chasing?

How can I prove $A \setminus(B \cup C) = (A \setminus B) \cap (A \setminus C)$ using element chasing? I need to verify that it is correct and show the steps of element chasing.
5
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0answers
75 views

Prove that there exists at least one root of $g$ between any two roots of $f$ [duplicate]

Given that $$f(x)= 1 - e^x\sin(x)$$ $$g(x)= 1 + e^x\cos(x)$$ Using Rolle's theorem, prove that there exists at least one root of $g$ between any two roots of $f$. Attempt so far: $f'(x) = ...
5
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4answers
82 views

An antonym for “converse”

Suppose you are proving $p \leftrightarrow q$. In your first paragraph you prove $p \rightarrow q$. Your second paragraph begins, “For the converse, assume $q$ holds.” In this situation, we have a ...
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1answer
32 views

The image of an injective function whose domain is a topological space also a topology

Let $(X, T )$ be a topological space, and let $f : X → Y$ be an injective (but not necessarily surjective) function. QUESTIONS. (1) Is $T_f := \{ f(U) : U ∈ T \}$ necessarily a topology on $Y$ ? ...
2
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3answers
86 views

Using $\epsilon-\delta$ proof to prove continuity

Use an $\epsilon-\delta$ proof to show that $f : R \setminus \left \{ \frac{-3}{2} \right \} \rightarrow R$ , $$f(x) = \frac{3x^2-2x-5}{2x+3}$$ is continuous at $x = -1$ Hello there. Can ...
0
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1answer
18 views

Proof using mean value theorem

Prove using the mean value theorem that $e^{x+1}\geq 2e^x$ by considering the interval $[x,x+1]$. Using the definition, there exists a $c$ in $(x,x+1)$ such that $e^{x+1} - e^x = e^c$ (this is of ...