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

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11 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: ...
2
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1answer
26 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
24 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
30 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
28 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
20 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
37 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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3answers
42 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
49 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 ...
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2answers
228 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
15 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
38 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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0answers
20 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
20 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
14 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 ...
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2answers
37 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
29 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
38 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 ...
0
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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
42 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
37 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
30 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
56 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 ...
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2answers
29 views

Is this a valid method of proof?

We are given that $y = a + b$, and we want to prove that $y = a + c$ (using all the usual properties of numbers that we know from grade school). Does it suffice to set $a + b = a + c$, and by ...
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2answers
17 views

Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.

I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain. let $a,b\in R$ s.t. $ab\in P$ , ...
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0answers
28 views

Applying Stone Weierstrass to this isometry of $C^\ast$-algebra

I proved the following theorem but I'd like to confirm the last part of my proof. Statement: Let $A$ be a non-zero commutative $C^\ast$ algebra. Then $\varphi : A \to C_0 (\Omega(A))$ defined by $a ...
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5answers
64 views

Guessing on the SATs, is it ever better to leave it blank than to guess?

On most SAT questions, there are 5 answers of which exactly one is correct and exactly four are wrong. If one answers correctly you get $1$ point. If you answer incorrectly, you receive $-\frac14$ ...
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2answers
61 views

Showing that ∀x ∈ S ∀y ∈ E ( Q(x, y) → R(x) ) ≡ ∀x ∈ S (∃y ∈ E Q(x, y)) → R(x)

Q(x, y) := “Student x did exercise y in the book” R(x) := “Student x gets an A in the class” So my goal is to show that the following equivalency holds: ∀x ∈ S ∀y ∈ E ( Q(x, y) → R(x) ) ≡ ∀x ∈ S ...
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4answers
27 views

Proving a increasing function with algebra

I'm attempting to prove a quadratic function is increasing without any calculus, just using algebra facts. My question: Consider the function $g(x) = (x + \dfrac{1}{2})^2 + \dfrac{7}{4}$ Prove that ...
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0answers
16 views

A question involving Partial Steiner Triple Systems

I've been given the following question, which I think I've completed, but I just wanted to check whether what I've said is valid. Suppose that a PSTS(23) with a $K_5$ leave is constructed using ...
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2answers
32 views

Proof by induction for $ 2^{k + 1} - 1 > 2k^2 + 2k + 1$ for $k > 4$

I was given this proof for hw. Prove that $ 2^{k + 1} - 1 > 2k^2 + 2k + 1$ So, far I've gotten this Basis: $k = 5$, $2^{5 + 1} - 1 > 2\cdot5^2 + 2\cdot5 + 1$ => $63 > 61$ (So, the basis ...
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2answers
13 views

Variance of sample mean (problems with proof)

Assuming that I have $\{x_1,\ldots, x_N\}$ - an iid (independent identically distributed) sample size $N$ of observations of random variable $\xi$ with unknown mean $m_1$, variance (second central ...
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1answer
36 views

Prove that all subsequential limits are contained within a closed interval

Let $a, b$ be two real numbers such that $a < b$, and suppose that $(s_n)_{n=1}^\infty$ is a sequence such that $\forall\,\, n\,\, a \leq s_n \leq b$. Prove that all subsequential limits are ...
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1answer
85 views

Proving a strange identity

Numerically, it would seem the following identity holds true: $$\frac{6}{7}=\lim_{n\to\infty}\sqrt[n]{\sum_{k=3}^\infty{\left(k-\sum_{j=1}^{k}\frac{1}{j}\right)^{-n}}}$$ Down below I have proven ...
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0answers
26 views

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime.

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime. Assume $a^m$ has order $n$ and, $m$ and $n$ are not relatively prime. Then ...
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0answers
16 views

Show that $P_A(\cdot):=P(\cdot \mid A)$ is a probability measure

Let $(\Omega,\mathcal{A},P)$ be a probability space. (i) Show that if $P(A)>0$, then $$ P_A(\cdot):=P(\cdot \mid A) $$ is a probability measure on $(\Omega,\mathcal{A})$. (ii) Is ...
0
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3answers
34 views

Uniform convergence of $f_n(x) = n \sin(\frac{x}{n}) , x \in [-r,r]$

It is asked to prove that $$f_n(x) = n \sin(\frac{x}{n}) , x \in [-r,r]$$ Converges uniformly on the given interval for $r>0.$ The resolution of this suggested considered the fact that the ...
3
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1answer
48 views

Show that $f$ is continuous if it follows the intermediate value property

If $f: [a,b] \to \mathbb{R}$ is $1-1$ and has the intermediate-value property -- that is, if $y$ is between $f(u)$ and $f(v)$, there is at least one $x$ between $u$ and $v$ such that $f(x)=y$ -- show ...
0
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1answer
54 views

Spotting mistake: unnecessary given condition

I have solved the following problem without using a given premise. Could someone please spot whether I have done something wrong? Suppose we have a relation $\geq$ that is transitive, but not ...
0
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0answers
13 views

Show that $\mathbb{N}=\{ 0\} \sqcup S(n)$ and also that that union is disjoint?

Show that $\mathbb{N}=\{ 0\} \sqcup S(n)$ and also that that union is disjoint? I've been assigned this exercise in my lectures of elements of mathematics 2. Three axioms have been given for a Peano ...
0
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1answer
29 views

If $A$ is a subset of $C$ and $B$ is a subset of $C$, then the union of $A$ and $B$ is a subset of $C$

If $A$ is a subset of $C$ and $B$ is a subset of $C$, then $A\cup B$ is a subset of $C$. I was considering letting $x$ be an element of $A$ and $B$ and going from there, but I'm not sure that that is ...