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

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2answers
35 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) + ...
2
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2answers
23 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 $n = 2$. $2^{k + 1} \le 3^k$ ...
0
votes
0answers
7 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} ...
4
votes
1answer
40 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
19 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
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2answers
27 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
27 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
26 views

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

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
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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
29 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, ...
3
votes
1answer
46 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
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0answers
21 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
8 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 ...
1
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1answer
30 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 ...
2
votes
5answers
84 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
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
29 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: ...
3
votes
1answer
36 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 ...
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
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1answer
45 views

Finding residues at a point $a$ where $a$ is a pole.

I am faced with the following problem: (a) Find $\displaystyle res_{a} \frac{\varphi(z)}{(z-a)^{n}}$ where $\varphi$ is a given function analytic at $a$, $\varphi(a) \neq 0$, and $n$ is a positive ...
1
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1answer
27 views

Let $A$ a Noetherian ring and $ q \in\mathrm{Spec} (A)$. Then $q^{(n)} A_q=q^{n}A_q$.

Let $A$ be a Noetherian ring and $ q \in\mathrm{Spec} (A)$. Then $q^{(n)} A_q=q^{n}A_q$, where $q^{(n)}= \lbrace a \in A \mid \exists d \in A \setminus q\text{ such that }da \in q^n \rbrace$ and ...
-2
votes
0answers
26 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 ...
1
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1answer
610 views

Proving the Nested Interval Property using Axiom of Completeness

I'm self-studying real analysis using Abbott's text "Understanding Analysis." I'm trying to think out/prove as much on my own as I can, so I am working on proving the Nested Interval Property (Theorem ...
0
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0answers
16 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 ...
2
votes
1answer
56 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 ...
1
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0answers
31 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 ...
0
votes
1answer
491 views

Prove If a set contains more vectors than there are entries in each vector, then the set is linearly dependent

I want to prove this theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set $\{ v_1,v_2,...,v_p \}$ in $\mathbb{R}^n$ is ...
3
votes
2answers
26 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
10 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
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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 ...
1
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1answer
13 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 ...
0
votes
2answers
36 views

Prove that if A and B are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$

Prove that if A and B are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$. It was suggested to me that the easiest way to approach this was with a proof by ...
2
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0answers
45 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 ...
3
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3answers
31 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 ...
0
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0answers
32 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
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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)\} ...
0
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1answer
52 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 ...
1
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1answer
22 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.
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
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 ...
1
vote
4answers
35 views

Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...
0
votes
2answers
36 views

Is giving an explicit homeomorphism sufficient to prove that there exists a homeomorphism?

I am asked to prove that a square is homeomorphic to a circle. Now we can construct the homeomorphism explicitly by first having a bijection $\gamma$ that takes an arbitrary square in $\mathbb{R}^2$ ...
1
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0answers
11 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 ...
2
votes
0answers
32 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 + ...
3
votes
0answers
41 views

If $\{T_n\} \to T$ and $\{u_n\} \to u$, then $\{T_n(u_n)\} \to T(u)$.

Let $X$ and $Y$ be normed linear spaces. Define $$L(X,Y) = \{T:X \to Y \ \big | \ T \text{ is bounded}\}.$$ Let $\{T_n\} \to T$ in $L(X,Y)$ and $\{u_n\} \to u$ in $X$, then $\{T_n(u_n)\} \to T(u)$ ...
1
vote
1answer
15 views

Given (A => B) => C , B, prove C with resolution method

So with the premises $(A \Rightarrow B) \Rightarrow C$ $B$ It is easy to prove $C$ in the Fitch method, as in the proof below proof Therefore I should be able to prove it using a resolution proof ...
0
votes
0answers
17 views

Let $S$ be an infinite set. Then there is a set $A\subset S$ such that $A\sim \mathbb{N}$. [duplicate]

This proof seems to follow very quickly from the definition of infinite sets, but I feel like what I have is incomplete: Since $S$ is infinite, we have that $S\sim \mathbb{N}$. By definition, $S$ is ...