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

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28 views

some combinatorial proofs

These were simple induction proofs, so I decided to try and prove them combinatorially. I think I nailed the first one, not so sure about the second one. $\sum_{i=1}^n(i)(i!)=(n+1)!-1$ Have $n+1$ ...
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0answers
7 views

Show that the closed classes are the maximal elements of the partial order

In the lecture we defined a partial order $\leq$ on the communicating classes associated to a Markov chain. Now it is to show that the maximal elements of the partial order $\leq$ are the ...
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0answers
10 views

Closed communicating class and stochastic matrix

Let $(X_n)_{n\in\mathbb{N}_0}$ be a Markov chain with state space $E$ and transition matrix $(p_{ij})_{i,j\in E}$. Let $C\subseteq E$ be a closed communicating class. Show that $$ ...
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1answer
36 views

Proving that $F(x)$ is a constant

This was on a test and i know i was supposed to use 2nd ftoc to prove that $F(x)$ was a constant when $x>0$ $$ F(x) = \int_{0}^{x} \frac{1}{t^2 +1} dt + \int_{0}^{\frac{1}{x}} \frac{1}{t^2 +1} ...
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1answer
20 views

Confusion about Concavity

Simple question. The function g of a single variable is defined by g(x) = f(ax + b), where f is a concave function of a single variable that is not necessarily differentiable, and a and b are ...
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0answers
21 views

Proof check of sum of a compact and closed set of real numbers is closed

Let $A$ be a closed and $B$ be a closed and bounded set in $\mathbb R$ , then we have to show that $A+B:=\{a+b:a\in A , b\in B \}$ is closed in $\mathbb R$ . My Proof : Let $\{a_n+b_n\}$ be a ...
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1answer
27 views

characteristic function differentiation

Let $\mu$ be a probability measure on $\mathbb{R}$. Then the characteristic function is: $$ \varphi: \mathbb{R} \rightarrow \mathbb{C} \;\;\ \varphi(t):=i\int_\mathbb{R} e^{itx}d\mu(x) $$ Prove with ...
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1answer
42 views

The probability that $3$ random points on the circumference form a right-angled triangle?

In my probability theory course, I dealt with a similar problem which asks for the probability that $3$ random points on the circumference of a circle lie on the same semi-circle. But it makes me ...
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1answer
22 views

Proof verification for $fgh=1_A\dots\implies f,g,h$ are all bijections. - Cohn - Classic Algebra Page 15

Is the proof below correct? Thank you for your time! Notation: $xfgh\equiv h(g(f(x)))= (h \circ g \circ f)(x)$ Theorem: If $f:A\to B, g:B\to C, h:C\to A$ are three mappings such that $fgh=1_A$, ...
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2answers
23 views

Lebesgue Measure: No Atoms!

Disclaimer: This is just meant as record of a proof. For more details see: Answer own Question How to prove that the Lebesgue measure has no atoms: $$\lambda:\mathbb{R}^n\to\mathbb{R}_+$$ (Recall ...
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1answer
34 views

Flaw in this proof that the union of two open sets is open?

I'm trying to show that if U, V are open sets of $\mathbb{C}$, then U $\cup$ V is an open set of $\mathbb{C}$. My attempt at proving this is as follows: If $U$ is open, $\forall x$ $\in$ U, ...
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0answers
36 views

proving Q is dense in R

Definition: A set E is dense in X if every point x $\in$ $X$ is in $E$ or if every point of X is a limit point of E, or both. This seems automatically true if E is a subset of X. Though I can't ...
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0answers
15 views

Cofinite Topology: Borel Algebra?

Given the cofinite topology: $$\mathcal{T}:=\{U\subseteq\Omega:\#U^c<\infty\}$$ and generate its Borel algebra: $$\sigma(\mathcal{T})=\{E\subseteq\Omega:\#E\leq\aleph_0\lor\#E^c\leq\aleph_0\}$$ Why ...
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1answer
40 views

Proving these series equations to be equal?

I was recently attempting to prove the formulae which calculate the sum of arithmetic sequences where the difference between each term is just 1. I arrived at this formula first, which calculates the ...
2
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0answers
20 views

Set $E$ which halves the measure of an open interval [duplicate]

This was an exam question. I know that my answer is wrong, but I believe myself to be on the right track. Can someone help me finish my construction? Here is the question. Find a set $E$ with the ...
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1answer
26 views

Prove using anxioms that $1 \neq 0$ - checking

Let $1=0$ we know that $x \cdot y=0$ if $x=0$ or $y=0$ so $1(1-1)=0 $ if $1=0$ or $1=1$ but $1=1$ is contradiction since we assume that $1=0$ so $1 \neq 0$ Is my prof correct ? If not how should it ...
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1answer
18 views

Composition of a function with a metric

Which of the following functions, $f: [0,\infty) \rightarrow [0,\infty)$, can be composed with a metric $d$ to get a new metric $f \circ d$: a)$\;f(x) = \begin{cases}0 & \text{if $x=0$} \\x+1 ...
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1answer
22 views

Disproving using a Constructive Proof

I cannot find the n to prove the negation for the following: Disprove (Prove the negation) of: For every positive integer n, $3^n + 2$ is prime The way in which I have written the negation is: ...
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0answers
18 views

Estimating upper bound

Let the following Cauchy Problem be $\displaystyle\cases{ y'(t)=f(t,y(t)) & \cr y(0)=\eta }$ for $t\in[0,T]$ Define the approximation $y_n$ of $y(t_n)$ as: ...
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1answer
29 views

Combinations from prime number of elements

Let $p$ be a prime and let $k$ be a natural number: Prove that for $k < p$, $\binom{p}{k}$ is divisible by $p$. My proof: The formula for $p$ choose $k$ is: $$\frac{p!}{k!(p-k)!}$$ Since the ...
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1answer
27 views

Proving that $(A^t)^t=A$

$(A^t)^t=A$ $(A^t)^t-A=0$ $(A^t)^t-A=A-A\rightarrow (A^t)^t=A$ Is this proof is valid or do I need to add more information to make it more clear?
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1answer
44 views

What can you say about $f$ and $g$ in the case that $fg$ is 1) Injective, 2) Surjective - Cohn - Classic Algebra Page 15

Question: Are my proofs below valid? In both cases we are using: $f:A\to B, g: B\to C$ Notation of your type converted: $(g\circ f)(x)=g(f(x))=xfg$ If $fg$ is injective what can be said about ...
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3answers
37 views

Is this formula satisfiable?

I am confused whether or not my explanation for whether or not this formula is satisfiable is correct. Note that the question state it should be Brief and it should not be necessary to write down a ...
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0answers
21 views

Simplifying Gamma functions yet having a complication while graphing when the function was able to be graphed previous to simplification?

According to the Euler's duplication formula: $$ \Gamma(z) \Gamma(z+\frac{1}{2}) = 2^{1-2z} \sqrt{\pi} \Gamma(2z) \therefore $$ $$ \Gamma(2z) = \frac{\Gamma(z) \Gamma(z+\frac{1}{2}) ...
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1answer
38 views

Would you accept this proof for $(A^c)^c = A$?

In my exercises I had the following question: Prove that $(A^c)^c = A$. My solution: Let $A$ be a set where $A\subset X$. $A = \{x \in X, x \in A\}$ by definition. $A^c = \{x \in X, x \notin A\}$ ...
2
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1answer
51 views

Prove that $\det(A) > 0$

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a real $n \times n$ matrix such that : $A^{3} = A + \mathrm{I}_{n}$. Prove that $\det(A) > 0$. Here is what I tried : $X^{3}-X-1$ is a null polynomial ...
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2answers
46 views

How can I find fifth root of unity?

I have no idea to do this question, how can I find the fifth root of unity? Question : Find all the distinct fifth root of unity. Let $\alpha$ be a fifth root of unity such that $\alpha \ne 1$. ...
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2answers
54 views

The product of uniformly continuous functions is not necessarily uniformly continuous

I was asked to show that given two functions $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ which are both uniformly continuous, to show that the product ...
7
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2answers
229 views

Proving a palindromic integer with an even number of digits is divisible by 11

I'm in an introductory course for discrete math so I'm a novice at English proofs. I'm not sure if my reasoning here is valid or if I'm using modular arithmetic correctly. Specifically the line I ...
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1answer
17 views

Proof verification for $f$ & $g$ surjective implies $fg$ surjective - Cohn - Classic Algebra Page 15

Question: Is this a valid proof? Side question: Am I less likely to get answers based on using notation $xfg=g(f(x))$? I want to prove that if $f$ and $g$ are surjective, then $fg$ is ...
3
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2answers
41 views

Proof verification of compactness

Let $K$ be the set $\{0\} \cup \{1/n : n \text{ is an element of the positive integers}\} $ Prove that $K$ is compact. In my head, it seems that what they are asking in this question to prove is ...
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4answers
86 views

Algebraic Proofs in Combinatotics

Prove the following identity using an Algebraic Proof. $$\binom{n + m}{2} = nm + \binom{n}{2} + \binom{m}{2}$$ I have no idea where to begin on this problem or let alone finish it.
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22 views

Game of writing a binary sequence proof

Let $n \gt 2$ be a natural number. We consider the following game. Two players write a sequence of $0$s and $1$s. They start with an empty line and alternate their moves. In each move, a player writes ...
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2answers
21 views

If $G$ is simple, then $\epsilon \leq {v \choose 2}$ - Bondy/Murty - Graph Theory with Applications Page 4

Question: Does this proof hold? Is this a bad proof? Any nicer proofs that don't rely on other theorems? Notation: $\epsilon$ - Number of edges $v$ - Number of vertices G - Here, any Graph ...
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1answer
17 views

The number of ways to paint a red tile in a grid.

here's the question: "You have nine tiles arranged into a three by three square mosaic. If you color each tile red or blue with equal probability, what is the probability that there exists a two by ...
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1answer
29 views

Verification of a delta/epsilon Proof of continuity

So I am asked to show that $f:\mathbb R\rightarrow \mathbb R$ is strictly increasing and $f^{-1}: f(\mathbb R)\rightarrow \mathbb R$ is continuous at $1$. My $f(x)$ is a point wise function $$f(x) = ...
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0answers
15 views

Use induction and Pascal’s identity to prove that if n > 1, then 1 = n − 1 = n [on hold]

Use induction and Pascal’s identity to prove that if n > 1, then (n)C(1) = nC(n − 1) = n
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2answers
28 views

Induction on the number of marbles in a heap.

Here is the problem in full: "A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more ...
2
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2answers
38 views

Epsilon delta proof verification

Prove that $ \lim_{x \to a} 5x^3$exists for every $a \in \mathbb{R}$. Here's my proof. I was wondering if it is complete and notationally correct: Suppose $\epsilon > 0$ has been provided. ...
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3answers
52 views

Prove that $\sqrt[4]{1+y^4} \leq 1+|y|$

Prove that $\sqrt[4]{1+y^4} \leq 1+|y|$ for all real values of $y$. I attempted to show this by finding the power series expansion of $\sqrt[4]{1+y^4} $ and then relating that to $1+|y|$; however, I ...
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1answer
19 views

Prove that the linear map of the basis $V$ is a spanning set of the image of $f$

Suppose that $f:V\rightarrow W$ is a linear map of finite-dimensional vector spaces and that $S=\{v_1,v_2,...,v_n\}$ is a basis for $V$. Prove that $\{f(v_1),f(v_2),...,f(v_n)$} is a spanning set ...
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2answers
30 views

If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

Let $n\in\mathbb{N}$. So far I have: If the sum of the digits of $n$ is $k$, then $n = 9m + k$, where $m$ element of an integer (not sure why). Now consider $5n-n$. Help?
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3answers
39 views

If d is a norm on V, is $\frac{d(x,y)}{1+d(x,y)}$ a norm on V?

Let d be a norm on a vector space V and let $\psi:V \to [0,\infty)$ be a function defined as $\psi(v)=\frac{d(v)}{1+d(v)}$. Is $\psi$ a norm on $V$? It seems that $\psi$ does not satisfy the ...
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3answers
18 views

In $S_3$, determine the set $T=\{ x\in S_3 | x^2=e\}$. Is $T$ a subgroup of $S_3$?

Here's my solution: Is it right or wrong? $S_3=\{ \begin{cases} 1\mapsto1 \\ 2\mapsto 2 \\ 3\mapsto 3 \end{cases}, \begin{cases} 1\mapsto 2 \\ 2\mapsto 1 \\ 3\mapsto 3\end{cases}, \begin{cases} ...
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3answers
32 views

The second derivative of $f^{-1}$ and another question. :)

Suppose both $f$ and $f^{-1}$ are twice differentiable functions. Derive a formula for $(f^{-1})''$. My attempt: We have that by the inverse function theorem that: ...
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1answer
44 views

Use Fundamental Theorem of Arithmetic to prove that if $a >1$, $p$ is prime, and $p|a ^n$ for some $n \in \mathbb{N}$, then $p|a$

So, by the FTOA, since $a >1$, then a can be broken down into a product of a prime factors, so $a = p_1 \times p_2 \times \dotsm \times p_k$. Then, can I say that since $a$ is multiplied by itself ...
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1answer
38 views

Soundness of a simple tree edge count proof by induction

I'm trying practice and get better at proofs. Here is my attempt at a proof of the following simple statement: There are $n-1$ edges in a $n$ vertex tree. We will prove this by induction on $n$ ...
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1answer
31 views

Verify proof that ${p \choose r} ≡ 0 \pmod p$

Let $p$ be a prime number. For any $1 ≤ r ≤ p − 1$, prove that $${p \choose r} ≡ 0 \pmod p$$ I'm thinking that it suffices to show $p$ divides ${p \choose r}$. So then: $$\begin{align} p\ |\ {p ...
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1answer
16 views

Linear functionals and integration verification

Can you please verify my reasoning? (a) Yes as (b) No, as function is squared (c) Yes, same reasoning as (a), squared values of x do not affect linearity. Does the region of integration affect ...
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0answers
67 views
+50

About a result concerning Mersenne primes

I want to verify the proof of this result Theorem: If $p>2$ is a prime and $$H_{p}=\frac{(\sqrt3+2)^{2^{p-1}}+1}{(2^{p}-1)(\sqrt 3+2)^{2^{p-2}}}$$ is a natural number then $2^{p}-1$ is a prime ...