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

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2
votes
2answers
34 views

Can$A \cap (B' \cap C')$ be $(A \cap B') \cap (A \cap C')$?

If I use the above statement, provided that it is right, in a question, would I have to prove it as well?
2
votes
1answer
26 views

Interesting and unusual word problem with prime numbers and factors [on hold]

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 prime numbers, but other than that, the textbook gave no hints really and ...
2
votes
1answer
453 views

Show $\nabla^2g=-f$ almost everywhere

let a continuous function $f(x,y,z)$ be absolutely continuous over every bounded region and let it be in $L^1$ that is $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} ...
1
vote
1answer
14 views

The bases for the discrete topology

The collection $\mathcal{B} = \{ \{x\} : x \in X \}$ is a basis for the discrete topology on a set X. If X is a finite set with n elements, then clearly $\mathcal{B}$ also has n elements. Is there a ...
0
votes
0answers
8 views

Exploring the properties of the Srogenfrey Line

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 ...
5
votes
2answers
540 views

How do I prove the completeness of $\ell^p$?

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
0
votes
1answer
20 views

how to find all values satisfing a function whose depends on another function?

How can I find all values $x>0$ such that $\int_0^x [t]^2 \, \mathrm{d}t=2(x-1)$? Does there exist an analytic solution to this problem? I mean a non iterative method to find these values, where ...
0
votes
0answers
14 views

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 ...
0
votes
0answers
30 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}| ...
1
vote
4answers
49 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 ...
0
votes
3answers
25 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 ...
0
votes
1answer
20 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. ...
1
vote
2answers
37 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, ...
0
votes
1answer
804 views

Disjunctive normal form (BOTH dnf and cnf) example help

T or ( not x and y ) or ( x and y ) In class we went over examples of how many things can be both DNF and CNF... eg. (not z) or y can be thought of as both (not z) or y .... dnf ((not z) or ...
1
vote
1answer
18 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 ...
-1
votes
0answers
33 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 $a$.
-4
votes
0answers
34 views

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 $\{a_k\}_{k=1}^\infty$, $1>a_k>0$, $\sum_{k=1}^\infty a_k=\infty$. ...
1
vote
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 ...
1
vote
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 ...
0
votes
1answer
47 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 ...
0
votes
1answer
28 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 ...
1
vote
2answers
28 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} ...
1
vote
2answers
805 views

A set $A \subseteq \mathbb{R}$ is closed if and only if every convergent sequence in $\mathbb{R}$ completely contained in A has its limit in A

Real analysis is a topic I'm unfamiliar with and I'm confused on how to write proofs on them. In order to prove that: A set $A \subseteq \mathbb{R}$ is closed (1) $\iff$ Every convergent sequence in ...
3
votes
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 ...
0
votes
0answers
42 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 ...
0
votes
2answers
24 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 ...
1
vote
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 ...
0
votes
2answers
149 views

Can some inequalities help to pin down an unique solution in a linear system of equations with infinite solutions?

I need to discuss the number of solutions of the following system of equations. Any help would be very appreciated. Consider the known parameters $a_1,...,a_4;d_1,d_2,d_3$ such that $0< a_i< ...
0
votes
2answers
34 views
3
votes
2answers
86 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 ...
0
votes
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 ...
1
vote
0answers
33 views

If minimal degree is at least $k$, then the graph contians a path of length at least $k$ [duplicate]

$G$ is a simple graph that consists of a vertex set $V(G) = \{v_1, v_2, ..., v_n\}$ and an edge set $E(G) = \{e_1, e_2, ..., e_m\}$ where each edge is an ordered pair of vertices. The edge $\{u,v\}$ ...
1
vote
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 = ...
2
votes
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. $$\lim_{x \to \infty} \dfrac{x^a}{x^b} = 0$$ ...
2
votes
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 ...
0
votes
1answer
18 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 ...
2
votes
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 ...
0
votes
0answers
32 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 ...
0
votes
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 ...
2
votes
0answers
42 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
votes
0answers
35 views

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

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 ...
1
vote
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 ...
1
vote
1answer
20 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 ...
0
votes
1answer
13 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 ...
27
votes
2answers
3k views

How would one be able to prove mathematically that $1+1 = 2$?

Is it possible to prove that $1+1 = 2$? Or rather, how would one prove this algebraically or mathematically?
4
votes
1answer
1k views

If $\lim f(x) = 0,$ then $\lim 1/|f(x)| = \infty.$

The Problem: Suppose that $f:D\to\Bbb R$, where $D$ is a subset of $\Bbb R$ and $a$ is an accumulation point of $D$, $\lim_{x\to a}f(x)=0$, and $f(x)\ne0$ for any $x$ in $D$ in some neighborhood ...
1
vote
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 ...
0
votes
1answer
36 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 ...
3
votes
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 ...
3
votes
1answer
772 views

Counting Elements and Their Inverses

The problem I am attempting to prove is the following: In any finite group $G$, the number of elements not equal to their own inverse is an even number. Caveat: I have had very limited experience ...