For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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0
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2answers
17 views

Showing $u_1, u_2, u_3$ is basis

Let $\{v_1, v_2, v_3\}$ be a basis for a vector space $V$. I want to show that $\{u1, u2, u3\}$ is also a basis where $u1 = v1, u2 = v1 + v2$ and $u3 = v1 + v2 + v3$ I wanted to use the standard ...
2
votes
1answer
33 views

Problem understanding this specific proof that $\sqrt{2}$ is irrational.

The proof (taken from http://www.themathpage.com/aPreCalc/rational-irrational-numbers.htm#proof): "To prove that there is no rational number whose square is 2, suppose there were. Then we could ...
0
votes
1answer
19 views

Would it be correct to say that NOT(P OR Q) is (NOT P AND NOT Q)?

I seem to think it is true as $$ x \notin (A \cup B)$$ $$\implies x \notin A \text{ and }x\notin B$$ $$\implies x \in A^C \text{ and }x\in B^C$$ I have deduced this via a venn diagram, and ...
3
votes
4answers
424 views

Where is the problem wih this proof in complex numbers?

Our teacher gave us a hard question (according to her, it is pretty hard for our level): Given that $|z_1| = |z_2|= |z_3|=1,z \in\mathbb{C}$, prove that $|z_1+z_2+z_3| = ...
10
votes
3answers
57 views

Prove that $4x^2-8xy+5y^2\geq0$ - is this a valid proof?

I need to prove that $4x^2-8xy+5y^2\geq0$ holds for every real numbers $x, y$. First I start with another inequality, i.e. $4x^2-8xy+4y^2\geq0$, which clearly holds as it can be factorized into ...
1
vote
1answer
31 views

Induction Clarification

I had this problem: Is it always necessary to go from n to (n + 1) or from (n - 1) to n in the inductive hypothesis? Is the "direction" always important? Here is my solution to one such proof, which ...
2
votes
1answer
30 views

Verification of Solution for Walter Rudin Principles of Mathematical Analysis Exercise 20, Chapter 3

I have written an answer for the problem 20, chapter 3 of Walter Rudin's Principle of Mathematical Analysis. I think the proof is correct, but since I am new with this kind of proofs, I am skeptical ...
1
vote
1answer
13 views

U uniform on [-1,1] - Find density of U^2

Let $U$ be uniformly distributed on $[-1,1]$. Find the denstiy of $U^2$. I would start with $F_{U^2}(u)$=$P(U^2\le u)$=$P(-\sqrt{u}\le U\le\sqrt{u})$ for $u\ge 0$. Since it is uniformly distributed ...
1
vote
0answers
16 views

Proving that a union of a countable and an uncountable set is equivalent to the uncountable set (proof check)

Let $A$, $B$ be sets with $|B|=\aleph _0$ and $|A|>\aleph _0$, Prove that $|A\cup B| = |A|$ I've already seen somewhere here (though can't seem to find it now) a proof using the fact that ...
1
vote
1answer
30 views

Find the curve which together with $\gamma$ encloses the greatest area.

I'm working through Gelfand & Fomin's Calculus of Variations by myself, and could use the guidance of someone familiar with the subject. The problem I'm on now is the following: "Given two points ...
1
vote
0answers
24 views

Finite dense subset implies $X$ finite

Suppose $E \subset X$ is a finite dense subset. Prove that $X$ must also be finite. This is proven quite easily by showing that $\bar{E} = E$ since $E' = \emptyset$, so that $\bar{E} = X$. ...
0
votes
1answer
15 views

Completely Regular Spaces and Embeddings

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. We were going over separation axioms in class when assigned the following problem. Given ...
0
votes
1answer
30 views

Show that is $\upsilon$ is an eigenvector of the matrices A and AB

(assume invertibility) Show that is $\upsilon$ is an eigenvector of the matrices A and AB with corresponding eigenvalues $\lambda \neq 0$, $\mu$ respectively, then $\upsilon$ is also a corresponding ...
1
vote
1answer
16 views

Given that $f:V\rightarrow W$, prove that $v_1,..,v_n$ are linearly indipendent

Let $V , W$ be two vector spaces and $f : V \rightarrow W$ a linear map. Let $w_1,...,w_n$ be elements of $W$ that are linearly independent and let $v_1,...,v_n$ be elements of V such that $f(v_i) = ...
5
votes
2answers
2k views

Is this a valid attempt at the Riemann Hypothesis? [on hold]

From Marcus Du Sautoy's book “The music of the primes”, there is a method of finding a very long list of N consecutive numbers which are not primes. e.g $101!+2, 101!+3,...,101!+101$ all of which are ...
0
votes
1answer
8 views

Prove that the sequence is Cauchy

This is problem 17, page 55, from section 1.2: Cauchy Sequences in the textbook Introduction to Analysis, Fifth Edition, by Edward D. Gaughan. Prove that the sequence ...
1
vote
1answer
9 views

Pumping Lemma proof and the union/intersection of regular and non-regular languages

I am still learning the pumping lemma. I have a problem for which I used it. I used it on the first part (a) but I am unsure if it is correct. Parts b-d, I am not sure how to do it. I created a dfa ...
1
vote
1answer
59 views

Convergence of a sequence by convergence of sub-subsequence

Suppose that $\{p_n\}_{n \in \mathbb{N}}$ is a sequence in a metric space $X$. Assuming that every subsequence of $\{p_n\}_{n \in \mathbb{N}}$ has itself a subsequence that converges, say, to $p$, ...
0
votes
1answer
16 views

Using the Pumping Lemma To Prove A Language Is Not Regular

I am taking a Automata class and we just went over the Pumping Lemma. Initially, it did not make sense. I am still not fully comfortable but I have started trying to use it to prove that a language is ...
1
vote
1answer
24 views

Proof set theory involving instantiation

Is it okay to instantiate with the same element in universal and existential instantiation? Here follows my proof of the following theorem. Theorem If $A \subseteq B \setminus C $ and $A \not = ...
1
vote
2answers
34 views

$A\subseteq B\to C\setminus B\subseteq C\setminus A\,$ — how to prove this?

Given $A \subseteq B $. Prove for every set $C, C\setminus B \subseteq C \setminus A $. Logical Argument: Given: $\forall x, x \in A \rightarrow x \in B $ Goal: $\forall C \forall x , x\in ...
3
votes
1answer
40 views

$\sum_{n=0}^\infty z^n = \prod_{m=0}^\infty \left(1+z^{2^m}\right)$

When reading Iwaniec and Kowalski's Analytic Number Theory, I came across the following "identity" on page 11 (the Amazon link has a free book preview which includes page 11): $$\sum_{n=0}^\infty z^n ...
0
votes
2answers
19 views

Let $\text{A}$ be a nonsingular $\textit{n}\times\textit{n}$ matrix, and let $\textit{B}$ be a basis for $\mathbb{R}^n$

Show that $ B_1 = \{\textbf{Av}| \textbf{v} \in B\} $ is also a basis for $\mathbb{R}^n.$ I apologize for my informality, but I would really like some feedback as to whether I am using the correct ...
1
vote
1answer
38 views

Prove there exists a $c$ such that $-f'(c)=\frac{f(c)}c$

Let $f: \Bbb{R} \longrightarrow \Bbb{R}$ be a continuous function on $[0,2]$ and differentiable on $(0,2)$. $f(2)=0$. Prove there exists a $c \in (0,2)$ such that $-f'(c)=\frac{f(c)}c$. What I did: ...
0
votes
3answers
55 views

Show that $\sin \dfrac{n \pi}{4}$ is divergent.

Show that $\sin \dfrac{n \pi}{4}$ is divergent. My attempt: Consider the subsequences $x_{4n}=\sin (n \pi)$, which converges to $0$, and $x_{8n+2}=\sin \dfrac{2(4n+1) \pi}{4}$, which converges ...
0
votes
0answers
8 views

Sign Of Permutation That Is Written As C Different Cycles

prove: if $\sigma\in S_n$ is a factorization of $c$ disjoint cycles so $Sgn(\sigma)=(1)^{n-c}$ We know the one cycle sign is $(-1)^{l-1}$ so $c$ of them is $(-1)^{l-1}\cdot ...
0
votes
2answers
15 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
1
vote
0answers
19 views

Apply the Euclidean algorithm to $2$ and $1 − 3i$ in the integers of $\mathbb Q[\sqrt {-1}]$

Question: I have to apply the Euclidean algorithm to $2$ and $1 − 3i$ in the integers of $\mathbb Q[\sqrt {-1}]$ My Solution: Since $N(2) = 4$ and $N(1–3i) = 10,$ we must start by dividing $1–3i$ ...
0
votes
0answers
29 views

Proving if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$

Prove that if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$. My attempt: Since $f$ is integrable then there's a sequence of partitions ...
0
votes
1answer
10 views

Distance between sets in $\mathbb{R}^p$

Let $A$ and $B$ be closed, bounded, disjunct subsets of $\mathbb{R}^p$ Now, this is not a metric, but define $\delta$ like this: $$ \delta = \inf V, $$ where $$ V = \{ \| a-b \| \mid a \in A \text{ ...
0
votes
0answers
6 views

Proof regarding representation of numbers

Prove directly that two different representations to the base k represent different integers. I cannot use the basis representation theorem, only the fact that $a_sk^s + a_{s-1}k^{s-1} + ... + a_0 = ...
0
votes
2answers
36 views

If $(a_n)$ is an increasing sequence, $(b_n)$ is a decreasing sequence, $a_n \leq b_n \forall n \in \mathbb N$. Prove $\lim a_n \leq \lim b_n$

If $(a_n)$ is an increasing sequence and $(b_n)$ is a decreasing sequence, with $a_n \leq b_n \forall n \in \mathbb N$. Prove that $\lim a_n \leq \lim b_n$ This is the closest thing I found on this ...
1
vote
2answers
28 views

Proving that a nonzero integer n has a unique representation

This is the first proof I've written. Can anyone give me advice? I don't know if its valid, or if there are ways to improve / other ways to do it: Prove that each nonzero integer may be uniquely ...
1
vote
1answer
55 views

Is this definition valid?

I am working on this problem: "Suppose $f:A\times A\rightarrow A$. A set $C \subseteq A$ is closed under $f$ if $\forall (x,y) \in C \times C(f(x,y) \in C)$. Now suppose $B \subseteq C $. The closure ...
1
vote
1answer
52 views

How many numbers smaller than one million, their sum of digits is at least 20?

How many numbers smaller than one million, their sum of digits is at least 20? My attempt: Since I don't know how to handle the "at least" part, I'll be using a complement: The general case is ...
0
votes
1answer
22 views

Check my logical argument for this proof.

if x is a real number $x \not =\ 1 $, then there exists y which is also a real number $ ((y+1) \div ( y-2) ) = x .$ Prove it's converse also. Logical Argument: given: $x \not = 1$ Goal: $ ...
0
votes
1answer
21 views

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$.

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$. My definition of closure is "Let $(X, \mathfrak T)$ be ...
1
vote
0answers
27 views

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find $\overline A$, int$(A)$, and bdry$(A)$.

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find closure of $A$ $(\overline A)$, interior of $A$ (int$(A)$), and boundary of $A$ (bdry$(A)$). $A$ ...
2
votes
2answers
33 views

Set theory (containing Power Set) Need Help in a proof

I am confirming whether my proof is correct or not and need help. If $ A \subseteq 2^A , $ then $ 2^A \subseteq 2^{2^A} $ Proof: Given: $ \forall x ($ $ x\in A \rightarrow \exists S $ where $ ...
1
vote
1answer
31 views

If $(x_n)$ is a convergent sequence, for any $\epsilon>0, \exists M$ such that $|x_n-y_n|<\epsilon$ for all $n \geq M$. Is $(y_n)$ convergent?

If $(x_n)$ is a convergent sequence and $(y_n)$ is such that for any $\epsilon>0, \exists M$ such that $|x_n-y_n|<\epsilon, \forall n \geq M$. Is $(y_n)$ convergent? My attempt: Let lim ...
0
votes
1answer
17 views

Give example of convergent and divergent sequences of positive numbers such that lim $x_n^{\frac1n}=1$

So basically I have to give examples where the root test doesn't work, right? For the convergent sequence, I'm taking $x_n=(1,1,1,\cdots)$, then lim $x_n^{\frac1n}=1$ Is there any non-constant ...
2
votes
1answer
32 views

Simple true/false statements about function composition

Given the functions $f,g,h$ from $\mathbb{R}$ to $\mathbb{R}$ I have to determine whether the following statements are true: "If $f \circ g$ is strictly increasing and $f$ is injective then $g$ is ...
2
votes
2answers
31 views

Give an example to show that convergence of $|x_n|$ does not imply the convergence of $x_n$

I'm taking $x_n=(-1)^n=(-1,1,-1,1,\cdots)$, which is divergent, but $|x_n|=(1,1,1,1,1,\cdots)$ converges to $1$. Is this example correct?
0
votes
0answers
7 views

If a set A is bounded from above, then the set of upper bouds M has minimum

I hope the title is clear, because I am italian and I study calculus exclusively from italian books. I had to prove this proposition refusing to look the book (because if I read the proof, I'll ...
1
vote
1answer
12 views

Question on the argument proving primary decomposition theorem

Lang - Algebra p.150, Lemma 7.6 Let $E$ be a torsion module of exponent $p^r(r\geq 1)$ for some prime element $p$. Let $x_1\in E$ be an element of period $p^r$. Let $\bar E = E/(x_1)$. Let ...
0
votes
1answer
17 views

Under What Conditions Is $f:M\rightarrow \mathbb{C}$ Where M Is the Set of 2x2 Matrices a Function and Not a Function?

I came across a problem that I thought was interesting. I attempted to solve the problem below, and I would be grateful if someone would check my logic in what follows. Let the set M of all 2 by 2 ...
0
votes
0answers
22 views

In the field $GF(p^n)$, prove that for every positive divisor d of n, $x^{p^n }-x$ has an irreducible factor over $GF(p)$ of degree d

In the field $GF(p^n)$, prove that for every positive divisor d of n, $x^{p^n }-x$ has an irreducible factor over $GF(p)$ of degree d Let d be any divisor of n. Then $GF(p^d)$ is a subfield of ...
1
vote
2answers
45 views

If a group G is isomorphic to H, prove that Aut(G) is isomorphic to Aut(H)

If a group G is isomorphic to H, prove that Aut(G) is isomorphic to Aut(H) Properties of Isomorphisms acting on groups: Suppose that $\phi$ is an isomorphism from a group G onto a group H, then: ...
3
votes
2answers
32 views

How to prove this Dirac Delta Function property?

How to prove the equation below, using Dirac Delta function properties? $$ \delta(x^2-m^2)=\frac{1}{2|w|}(\delta(x-w)+\delta(x+w)) $$ where $$ w^2=|x|^2+m^2 $$ I tried to show it using $$ ...
-2
votes
1answer
29 views

The Zeros Localization Theorem and the Extreme value Theorem

My intro to analysis book calls the following theorem The Zeros localization theorem: Theorem: Let $p(x)=x^n + a_{n-1}x^{n-1} +\dots+a_1 +a_0,\ x \in \mathbb R$ , be a polynomial. Then all the zeros ...