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

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1answer
39 views

Can't find solution to trigonometric equation, need help to understand why!

I am struggling with the solution of an equation and I think as well with a lack of understanding when there is a solution for such a trigonometric problem and when there will be infinitely many ...
0
votes
0answers
9 views

Prove finite nonempty set of real numbers has a largest element

Prove with induction that every finite nonempty set of real numbers has a largest element. Now this is my idea, (Please fix my notation where it is wrong:) Let $A=\left\{a_i\in \mathbb{R}:i\in ...
1
vote
1answer
46 views

Mathematical induction proof problem: $\sum_{i=1}^{n-1} i(i+1) = \frac{n(n+1)(n-1)}3$

I am having difficulty proving the inductive hypothesis $(k+1)$ for the following statement: $$\sum_{i=1}^{n-1} (i(i+1)) = \frac{(n)(n+1)(n-1)}{3}$$ This is what I have so far: $$(Step \ 1) ...
1
vote
0answers
15 views

Are the following proofs correct: Let $f: X \to Y$, $f$ injective with range $Y$, then show that $f^{-1}$ is an injective function

Let $f: X \to Y$, $f$ is a function and injective with range $Y$, then show that $f^{-1}$ is a function, 2. an injection, 3. is it a bijection? Proof: $f^{-1}$ is a function. ...
1
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0answers
10 views

composition of continous functions is continuous

Let X, Y and Z metric spaces, and $f:X \to Y$ and $g:f(X) \to Z$ continuous functions (where $f(X)$ is a subespace of Y). Need to show that $h(x)=g(f(x))$ is also continuous. To prove it, I used the ...
0
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3answers
21 views

Need help with this proof ∀x ∈ R[∃y ∈ R(x + y = yx) ↔ x ≠ 1]

Having a hard time proving this one. I can prove this with a contradiction in the (→) direction but I'm stuck on how to prove this in the (←) direction where x ≠ 1 is the given and ∀x ∈ R[∃y ∈ R(x + ...
1
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0answers
32 views

Proof check: a group $G$ with presentation $[a,b\mid ab=e]$ is isomorphic to $\mathbb{Z}$

Given that $ab=e$, we know $b =a^{-1}$. Since $G = \langle a,b\rangle$ this implies $\langle a,b\rangle=\langle a,a^{-1}\rangle=\langle a\rangle=G$. The presentation rewritten in terms of $a$ is ...
0
votes
1answer
25 views

Find the number of vertices n of the tree?

Suppose a tree has $n$ vertices where half of these vertices are of degree $2$, six are of degree $3$, and the remaining are leaves. Find the number of vertices $n$ of the tree. Please do not find the ...
1
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0answers
21 views

Show that if all row-sums of a square matrix $A$ are equal to $0$, then $A$ is singular [duplicate]

I need to show that if all row-sums of a square matrix $A$ are equal to $0$, then the matrix is singular. My idea was that to represent the situation, I can do as follows: $$A\vec{x} = \vec{0}$$ ...
0
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0answers
6 views

On double products about $n^{\mu(n)}$, the Prime Number Theorem and Stoltz's theorem

Combining the Prime Number Theorem and Apostol's Theorem 4.13, (Apostol, Introduction to Analytic Number Theory (Springer)) one can prove that ...
0
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0answers
11 views

Proof $\forall n \in \Bbb N$ that $2^n \cdot \prod_{i = 1}^{n} (2i-1)$ is divisible by $n!$

I'm trying to prove it by induction. $P(1)$ holds true. My inductive hypothesis is $n!\ |\ 2^n \frac {2n!} {2^n n!}$ which simplifies to $n!\ |\ \frac {2n!} {n!}$. Next $P(n+1)$: $$(n+1)!\ |\ 2^{n+1} ...
2
votes
2answers
31 views

Prove that if $n \in \mathbb{N}$ and $n \ge 2$, then $2^{n + 1} \le 3^n$.

Prove that if $n \in \mathbb{N}$ and $n \ge 2$, then $2^{n + 1} \le 3^n$. My method: If $n = 2$, $2^{n + 1} \le 3^n$ then $2^3 \le 3^2$ is $8 \le 9$, which holds for $n = 2$. $2^{k + 1} \le 3^k$ ...
4
votes
1answer
43 views

Would this be a valid proof for $\left| x + y \right| \geq \left| x \right| - \left| y \right|$

I wanted to check if this was a valid proof for considering whether $\left|x + y \right| \geq \left| x \right| - \left| y \right|$. My proof is as follows: Case 1: Assume $x > 0, y>0$,then ...
3
votes
1answer
31 views

Finding a joint probability mass function

I have to find the joint probability mass function (pmf) of (X,Y) for the following problem: Roll a die repeatedly until a five or six appears, and let X be the number of rolls before a five or six ...
1
vote
2answers
33 views

Optimization with a Probability

Imagine two points in $ℝ^2$ at $(-1, 0)$ and $(1, 0)$. You would like to walk from one point to the next in the shortest distance possible. However, there is a line segment coming from the origin to a ...
1
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2answers
27 views

Use induction to prove that $2^n \gt n^3$ for every integer $n \ge 10$.

Use induction to prove that $2^n \gt n^3$ for every integer $n \ge 10$. My method: If $n = 10$, $2^n \gt n^3$ where $2^{10} \gt 10^3$ which is equivalent to $1024 \gt 1000$, which holds for $n = ...
0
votes
0answers
35 views

Biggest number of teams with 16 wins in a tournament

Here is a problem from a math competition - the solution of which requires the enumeration of combinations. I am asking for affirmation of my solution. Twenty teams are in a round-robin tournament; ...
0
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0answers
29 views

Central limit theorem and the sequence with general term $e^{-n} ( 1+n+ \cdots + n^n/n!)$ [Proof check] [duplicate]

As an exercise I need to find the limit of the said sequence $$e^{-n} ( 1+n+ \cdots + n^n/n!)$$ using the toolkit of probability theory. Since no solution (only hints) is provided, I would appreciate ...
0
votes
0answers
18 views

Would this be considered a valid proof for $\forall r \in R$ if $0 < r < 1 $, then $\frac{1}{r(1-r)}\geq 4$

I did a proof of the following $\forall r \in R$ if $0 < r < 1 $, then $\frac{1}{r(1-r)}\geq 4$ using a proof by contra-positive, which was different from the direct proof that the solutions ...
1
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1answer
32 views

Find error in proof for $f(x) < g(x) \implies \lim_{x\to a}f(x) < \lim_{x\to a}g(x)$

I know that it is not true that $f(x) < g(x) \implies \lim_{x\to a}f(x) < \lim_{x\to a}g(x)$ A counter example could be $f(x) = 0$ $g(x) = |x|$ if $x\neq 0,\quad g(0) = 1$ $a=0$ However, ...
1
vote
2answers
37 views

Probability Proof about A and B

I have to formally prove that: $$P(A) = P(A\wedge \neg B) + P(A\wedge B)$$ so I did like this: $$P(A\wedge \neg B) + P(A\wedge B)$$ $$=P(A\wedge \neg B) + P(A)\cdot P(B)$$ $$=P(A)\cdot P(\neg B) + ...
1
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0answers
33 views

How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
0
votes
1answer
13 views

Runge Kutta error estimation

I am trying to solve a numerical analysis dealing with Runge Kutta methods. The problem is in solving the differential equation: $$\frac{d \vec{y}(x)}{dx} = \vec{F}(x,\vec{y}).$$ Defining the error ...
2
votes
5answers
90 views

Story proof for $\sum_{k=0}^n {n \choose k} = 2^n$ [duplicate]

I found a solution online that uses the Binomial Theorem. Is it possible to prove this without using that theorem?
1
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1answer
32 views

Cauchy but not rapidly Cauchy

I want to show that the sequence $\{\frac{(-1)^n}{n}\}$ is Cauchy but not rapidly Cauchy. Here is the work I done so far. I am curiously if I made any errors. Consider the normed linear space ...
1
vote
1answer
12 views

Let $H = \{2^m : m \in \mathbb{Z}\}$ & define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rationals by $a\mathbin{R}b$ iff $a/b \in H$.

Let $H = \{2^m : m \in \mathbb{Z}\}$ and define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rational numbers by $a\mathbin{R}b$ if and only if $a/b \in H$. Prove that $R$ is an equivalence ...
1
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0answers
25 views

Hyperbola equation proof

I've been trying to prove the canonical form of the hyperbola by myself. $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ I started from the statement that ...
-1
votes
0answers
30 views

Connection between prime numbers and transcendental numbers

I think there may be a strong connection between prime numbers and transcendental numbers. I am unable to prove what I have in mind by myself, so I am seeking help. My hypothetic theorem would be: ...
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0answers
27 views

Proving the value of an integral [on hold]

Prove that, $\frac1{\sqrt{2\pi t}}\int_{-\infty}^{\alpha}e^{-\frac{x^2}{2t}+\beta x}\,dx=e^{\frac{\beta^2t}2} N\left(\frac{\alpha-\beta t}{\sqrt t}\right),$ where $\alpha , \beta$ and t are positive ...
0
votes
0answers
17 views

How can I verify the following equality?

$$\int_0^{\infty}\frac{C\exp(-\frac{mx^2}{\Omega})}{\Omega^m}\frac{1}{\sqrt{2\pi}\lambda\Omega}\exp\left(-\frac{(\ln ...
0
votes
0answers
11 views

Let $A$ be a nonempty set. Prove that if $S$ and $R$ are equivalence relations on $A$, then $S \cup R$ is both reflexive and symmetric.

Let $A$ be a nonempty set. Prove that if $S$ and $R$ are equivalence relations on $A$, then $S \cup R$ is both reflexive and symmetric. My method: Let $x \in A$ be given. Then $x \in S$ or $x \in ...
1
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0answers
32 views

Why does this proof involving the FTOC “work”?

Let $h:\mathbb{R}\to \mathbb{R}$ be a continuous function and $f,g:\mathbb{R} \to \mathbb{R}$ differentiable on all of $\mathbb{R}$. Define $F(x) = \int_{f(x)}^{g(x)} h(t) dt$. Calculate the ...
3
votes
1answer
37 views

The matrix square root is not differentiable on the boundary of the manifold of positive semi-definite matrices?

$\newcommand{\psym}{\operatorname{P}_{\ge 0}}$ $\newcommand{\Sig }{\Sigma}$ Let $\psym$ denote the subset of symmetric positive semi-definite matrices. Let $S:\psym \setminus \{0\} \to \psym ...
3
votes
2answers
28 views

A relation $R$ is defined on $\mathbb{Z}$ by $aRb$ if and only if $2a + 2b\equiv 0\pmod 4$. Prove that $R$ is an equivalence relation.

A relation $R$ is defined on $\mathbb{Z}$ by $aRb$ if and only if $2a + 2b\equiv 0\pmod 4$. Prove that $R$ is an equivalence relation. My method: Let $a \in \mathbb{Z}$ be given. So, for any $a \in ...
0
votes
0answers
11 views

Interior of a cone is a cone?

I've read somewhere that the interior of a cone is once again a cone. By cone I mean a set $S$ with the property that $(\forall x \in S)(\forall \lambda \geq 0)\ \lambda x \in S$. However, if we ...
1
vote
1answer
18 views

If $R/P$ is an integral domain then $P\vartriangleleft R$ is prime. [duplicate]

Let $R$ be a ring and let $P$ be a proper ideal of $R$. If the quotient ring, $R/P$ is an integral domain then $P\vartriangleleft R$ is prime. For $x,y\in R$ we have $(x+P)(y+P)=xy+P\in ...
3
votes
1answer
55 views

Deriving the Normalization formula for Associated Legendre functions: Stage $2$ of $4$

The question that follows is a continuation of this previous Stage $1$ question needed as part of a derivation of the Associated Legendre Functions Normalization Formula: ...
1
vote
1answer
14 views

Prove that if $f : A \rightarrow B$ is a function, $D \subseteq A$, and $E \subseteq A$ then $f(D) - f(E) \subseteq f(D - E)$.

Prove that if $f : A \rightarrow B$ is a function, $D \subseteq A$, and $E \subseteq A$ then $f(D) - f(E) \subseteq f(D - E)$. My method: Let $y \in f(D) - f(E)$. Hence $y \in f(D)$ and $y \notin ...
2
votes
1answer
65 views

VERIFICATION: Prove that $\int_{-\infty}^{\infty}\frac{1-b+x^{2}}{\left(1-b+x^{2}\right)^{2}+4bx^{2}}dx=\pi$ for $0<b<1$

I need some reassurance that what I did here actually shows what need to be shown. Please correct me if I'm wrong. In Donald Sarason's "Notes on complex function theory", this question appears at ...
2
votes
0answers
46 views

Let $p$ be a prime. Let $f(x) = 3x+1$ and $g(x) = 6x+1$. Show that if $f(x) = p$, then $g(y) = p$. [duplicate]

The full question states: Let $p$ be a prime. Let $f(x) = 3x+1$ and $g(x) = 6x+1$. Show that: if there exists $x\in \Bbb N$ such that $f(x) = p$, then there exists $y\in \Bbb N$ $g(y) = p$. My ...
0
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0answers
37 views

USAMO 2005, Problem3 (Triangle Geometry)- Is my solution correct?

USAMO 2005, Problem 3: Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$. Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is ...
0
votes
1answer
12 views

On $\Bbb R^2$, Are unit circle centred at the origin and the origin homotopic equivalent?

I guess these two spaces are not homotopic equivalent. I suppose there are homotopic equivalent. Let $X=\{x \in \Bbb R^2 : ||x||=1\}$ $Y=\{(0,0)\}$ And there exists two functions $f: X\to \{(0,0)\} ...
3
votes
3answers
32 views

Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$

How do I show that $\ln(x) = \lim_{n\to\infty} n (x^{1/n} - 1)$? I ran into this identity on this stackoverflow question. I haven't been able to find any proof online and my efforts to get from ...
1
vote
1answer
23 views

If $g$ and $h$ are primitive roots of an odd prime $p$, then $g = h^k \pmod p$ for some integer $k$. Show that $k$ is odd.

If $g$ and $h$ are primitive roots of an odd prime $p$, then $g = h^k \pmod p$ for some integer $k$. Show that $k$ is odd.
0
votes
1answer
53 views

Show that $\{1/n:n∈N\}∪\{0\}$ is compact

The set is in $R^1$ and consists of $0$ and the numbers $1/n$. Call it $E$. Take a set of $n$ intervals of radius $r$, centered less than $2r$ apart and such that $\sum_{i=1}^n r \ge 1/2$. Call the ...
2
votes
3answers
28 views

Induction to prove that for any $r \in \mathbb{R}$ such tht $r \notin (0,1)$ $\sum_{i=1}^n r^i-1 = \frac{(1-r^n)}{1-r}$ for all $n \in \mathbb{N}$.

Use induction to prove that for any $r \in \mathbb{R}$ such that $r \notin (0,1)$ $$\sum_{i=1}^n r^{i-1} = \frac{1-r^n}{1-r}$$ for all $n \in \mathbb{N}$. My method: Assume $$\sum_{i=1}^k r^{i-1} = ...
1
vote
0answers
12 views

$3$ intersection points for a quartic polynomial implies 4 intersection points or a local extrema at one of the intersection points

Q: It is given that the graph of $y = x^4+ax^3+bx^2+cx+d$ (where $a,b,c,d$ are real) has at least $3$ points of intersection with the $x$-axis. Prove that either there are exactly $4$ distinct points ...
-1
votes
1answer
50 views

Set of rational numbers bounded between two irrationals is a closed set?

Consider the metric space $\mathbb{R}$ equipped with the standard distance metric. Let $S$ be a set of rational numbers in the open interval $(a,b)$ where $a$ and $b$ are irrational. Prove that $S$ is ...
2
votes
0answers
33 views

Proving that $f$ and $g$ are identically $0$ on the entire domain

Q: Let $f$ and $g$ be two non-decreasing twice differentiable functions defined on an interval $(a,b)$ such that for each $x\in (a,b)$, $f''(x)=g(x)$ and $g''(x)=f(x)$. Suppose also that $f(x)g(x)$ is ...
1
vote
0answers
23 views

Logistic model - solution verification

I'm looking at the Logistic model: $$\begin{cases} \dot{X} = X(1-X)\\ X(0) = X_0 \end{cases}$$ where the phase space is $M = \mathbb{R}$. The solution appears to be $X(t) = \dfrac{1}{1 + ...