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

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4
votes
3answers
57 views

In proving A = B, A, B are sets, do you always have to show $\subseteq$ and $\supseteq$?

I am trying to show the DeMorgan's Law $X \backslash \bigcup_{\alpha \in I} A_\alpha = \bigcap_{\alpha \in I} (X \backslash A_\alpha)$ It seems I could directly approach this as follows: $X ...
1
vote
1answer
29 views

Proof for: Let A and B be sets s.t $ A \cap B = A $ iff $ A \subseteq B $

I am practicing some proofs involving sets and I would like to see if what I did was a valid proof because it seemed to be different from the one provided in the textbook I am using given that it did ...
1
vote
5answers
88 views

Showing that $n! > n^2$ for $n\geq4$ by induction

My attempt: Prove $ n! > n^2 $ for $ n \geq 4 $ Base Case: $P(4) = 24 > 16$ Inductive Hypothesis $P(k) : k! > k^2 $ $P(k+1) : (k+1)! > (k+1)^2 $ $ (k + 1)! - (k+1)^2 > 0 $ $ ...
3
votes
1answer
55 views

Is this proof valid? The claim is $2^{k} < (k+1)!$ for $k \geq 2$

Hey guys so I think I have completed this proof but I'm not sure if its valid. Here it is: Prove that $ 2^n < (n+1)! \quad\text{for}\quad n >= 2 $ Here is my proof: Base Case P(2) = $ 4 < ...
1
vote
0answers
20 views

Witten Laplacian problem-show that $\lim\limits_{n \rightarrow +\infty} ||f_nu||=0$

Can someone help me please to correct my answer of this problem: We consider the Witten Laplacian with domain $C^∞_0(R^d)$. We know that this operator is essentially self adjoint,positive and with ...
5
votes
1answer
111 views

Proof verification : a very useful theorem (in measure theory)

Let $\bigcup_{n=1}^\infty E_n=E$ and $ E_{n} \subseteq E_{n+1} $ then $\lim\limits_{n\mapsto \infty} \mu^*(E_n) = \mu^*(E) $ even if each $E_n$ is a non-measurable set, where $\mu^*$ is outer ...
3
votes
2answers
57 views

Prove that if $2n+1$ and $3n+1$ are both perfect squares then $40|n$.

Prove that if $2n+1$ and $3n+1$ are both perfect squares then $40|n$. First, I took $$2n+1 \equiv x^2 \equiv 0, 1 \pmod 4$$ which showed that $n$ was even. Now, $$3n + 1 \equiv y^2 \equiv 0, 1, ...
-1
votes
1answer
28 views

Proof binomial coefficient [closed]

I'm trying to prove the following: $$\binom{n + p}{k} = \sum_{j=0}^n \binom{n}{j} \cdot \binom{p}{k - j}$$ How do I do it? Induction? And can someone hint me at how to start?
1
vote
1answer
17 views

Proving that $M \setminus {( A \cup B)} = (M \setminus A)\cap(M \setminus B)$

I want to prove that the above terms are logically equivalent. M is a set and $A,B\subseteq M$. $x \in M\setminus ( A\cup B ) $ $\Leftrightarrow x \in M \wedge x\notin A \cup B $ $\Leftrightarrow ...
1
vote
1answer
36 views

Is this proof for Theorem 16.4 Munkers Topology correct?

The followings is the Theorem 16.4 from Munkers' Topology: In the textbook it uses concept of subbasis to prove the theorem which I can't understand it. I tried to prove that in another way but I ...
0
votes
1answer
43 views

Iterative method for finding real solutions to $a+b+c+d = abcd = 7.11$

I have "come up with" a method for finding $a,b,c,d \in \Bbb{R}$ such that their sum and product is equal and wanted to ask if the method is sound. First, rearrange both equations so that only $a, b$ ...
0
votes
0answers
11 views

Prove that the number of nodes in ORBDD for fn with given order On is 2n +2

I cannot embed image yet, so I have no choice but to include a link here. (Also if anyone can include a link/tutorial/guide to how to display notations, I'd really appreciate it) This image contains ...
0
votes
1answer
47 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
12 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
61 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
20 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. ...
0
votes
3answers
22 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
vote
0answers
34 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
2answers
53 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
vote
0answers
23 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
votes
0answers
12 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
33 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
46 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
40 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 ...
2
votes
0answers
35 views

Gaussian process via RKHS construction: joint measurability comes for free?

Billingsley's "Probability and Measure" (and other books) show the joint measurability of the Brownian motion using the continuity of paths. Makes me wonder if we can say the joint measurability ...
1
vote
2answers
53 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
vote
2answers
28 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
42 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
votes
0answers
32 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
vote
1answer
37 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
44 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
vote
0answers
41 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. ...
1
vote
1answer
16 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 ...
3
votes
5answers
103 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
vote
1answer
39 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
13 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
vote
0answers
27 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
43 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: ...
0
votes
0answers
19 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
12 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
vote
0answers
35 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
39 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
29 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
12 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
21 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
60 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
15 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
71 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
47 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 ...