For questions about the formulation of a proof, not about the mathematics behind it.

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0
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0answers
64 views

Is it correct? $1^n +2^n +…+(p-1)^n=-1 \pmod p$

$p$ a prime number, $n\in \mathbb{N} $ and $p-1\mid n$ then $1^n +2^n +...+(p-1)^n=-1 \pmod p$ I'm not sure if my proof is correct: Take the group $G=(\mathbb{Z}^{*}_{p},\cdot)$ with the ...
1
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1answer
23 views

Which subsets of $l^2$ are compact?

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ State whether the following subsets ...
1
vote
1answer
10 views

Vertex deletion and chromatic number proof

Let G be a graph such that, for all vertices $a$ and $b$, $\chi(G-${$a-b$}$)=\chi(G)-2$. Prove that G is a complete graph. I started by drawing $K_5$ which has chromatic number $\chi(K_5)=5$ and ...
1
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1answer
37 views

Proof that every non-empty subset of a woset (X, $\leq$) has a unique minimal element.

I want to prove that every nonempty subset of a woset (X, $\leq$) has a unique minimal element. What I’m looking for: clarification and/or hints. I want to solve it on my own, but this is all the ...
0
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1answer
38 views

How to write this down in a rigorous manner?

I'm studying Quantum Mechanics and there's something I'm in doubt in how to write it down in a rigorous way. The idea is: consider a Hilbert space $\mathcal{H}$ and one hermitian operator $A\in ...
1
vote
2answers
936 views

Deriving Universal Modus Tollens

I'm asked to derive the validity of Universal Modus Tollens from the validity of Universal Instantiation and Modus Tollens. I'm new to this deriving/proving stuff, so I'm not sure if I'm doing it ...
0
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1answer
22 views

Under what conditions can we move the limit symbol through the logarithm symbol?

I was reading the derivation of the derivation of a log function. And saw this: $$\frac{d}{dx}[\log_b x]= \frac{1}{x}\lim_{v \to 0} [\log_b(1+v)^\frac{1}{v}]$$ Then, the limit notation gets moved ...
0
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0answers
34 views

How do I memorize mathematical proofs?

I first started wanting to know about the derivation of theorems because certain ones help you memorize the theorems better. But as I take harder math classes, it turns out better for me to use ...
0
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2answers
37 views

Algebraically transform logic expression

Algebraically transform: $\neg \forall x(P(x) \wedge Q(y) \implies \exists zR(z))$ to $\exists x\forall z(P(x) \wedge Q(y) \wedge \neg R(z))$ Justify each step with one or more ...
2
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2answers
2k views

Is the set of all invertible $n \times n$ matrices a vector space?

I'm studying Algebra and I'm asked to prove or disprove "Is the set of all invertible $n \times n$ matrices a vector space?" I assume with respect to the usual matrix-sum and scalar multiplication. I ...
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4answers
74 views

Find $A$ and $B$ such that $A⊈B$ and $B⊈A$? [on hold]

I need to prove that the subset relation “$⊆$” on all subsets of $\mathbb Z$ is not a total order and I'm going to do this by finding $A$ and $B$ such that $A⊈B$ and $B⊈A$? Is there a simple solution ...
0
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1answer
70 views

Estimating the Riemann integral of $f$ using an upper bound for $f$

Show is that the Riemann integral $\int_a^b f(x)\,dx$ is bounded by $M(b-a)$, where $M$ is a bound of $f(x)$. I was thinking I would show that $M(b-a)$ is larger than every Riemann sum, but my ...
2
votes
3answers
52 views

Proof with Fibonacci Sequence

I was working on my homework assignment for one of my classes and I have come across a proof question that my classmates and I are finding difficult to answer. The problem is asking us to prove that ...
3
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1answer
39 views

Show that $S_4$ is not isomorphic to $D_4\times\mathbb{Z}_3$

Show that $S_4$ is not isomorphic to $D_4\times\mathbb{Z}_3$ I have no idea how to show this. I'm studying for a test, so I am less interested in solutions and hints than I am strategy. What ...
1
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4answers
49 views

Details of proof by contradiction

I realize this is pretty basic but recently became unsure of how to justify proof by contradiction. Is it that case that I can show $A\Rightarrow B$ by assuming $A$ and NOT $B$, and showing this leads ...
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2answers
626 views

Proving the Division Algorithm using induction

Let $n \in \mathbb{N}$. For every $m \in \mathbb{Z}$, there exist unique $q, r \in \mathbb{Z}$ such that $ m = qn+r$ and $0 \le r \le n-1$. We call $q$ the quotient and $r$ the remainder when dividing ...
0
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2answers
42 views

If $X$ is a compact metric space and $E_n$ is closed nonempty subset, show that $\cap_{n=1}^\infty E_n$ is nonempty.

Suppose that $(X,d)$ is a compact metric space and $(E_n)$ is any sequence of nonempty closed subsets of $X$ with $E_{n+1}\subset E_n$ for all $n\in\mathbb{N}$. Show that $\cap_{n=1}^\infty E_n$ is ...
1
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1answer
34 views

Logical form of a set-theoretic statement.

From Velleman's 'How to Prove it' book, there is one statement - written below - of which I don't know how to write the logical form of, and I'm wondering if somebody could write it out. The ...
2
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1answer
475 views

Four-point geometry proof

I'm new to writing proofs and am working with proving finite geometry systems. I'm not sure how I should answer this one. Using the four point finite geometry system: prove that there exists a set of ...
1
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1answer
31 views

If $a$ is the only element of order $2$ in a group, show that $a$ is in the center of the group.

If $a$ is the only element with order $2$ in a group $G$, then $a \in Z(G)$. I'm studying for a test and I can't figure out how to prove it. What kind of methods might I try to solve this ...
0
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1answer
47 views

Subset relation ⊆ on all subsets of ℤ is a partial order, not a total order.

I need to prove that the subset relation “⊆” on all subsets of ℤ is a partial order but not a total order. I'm not experienced in these kind of proofs and was hoping to see an example of an easier one ...
1
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4answers
71 views

If $0\lt y \le 1$, prove that there exists a unique positive real number $x$ such that $x^2=y$

I'm stumped. I don't want an entire solution, just a hint. If $0\lt y \le 1$, prove that there exists a unique positive real number $x$ such that $x^2=y$ The section in the book I'm on is the least ...
4
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4answers
295 views

Definition of Inverse in Linear and Abstract Algebra

In a linear algebra text, the following is the definition of the inverse of a matrix An $n\times n$ matrix $A$ is invertible when there exists an $n \times n$ matrix $B$ such that $$AB = BA = ...
1
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3answers
39 views

How to prove that $x^2 + 3y^2 = 1$ is contained inside of the unit ball?

What is the best way to show that $S = \{(x,y) | x^2 + 3y^2 = 1\}$ is contained in the unit ball without graphing the set?
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7answers
72 views

Proving $\frac{n}{n+1} < \frac{n+1}{n+2}$ by induction?

I have the inequality $\frac{n}{n+1} < \frac{n+1}{n+2}$ I'm not sure how to go about proving it. I've started by testing with n = 1, which results in $\frac{1}{2} < \frac{2}{3}$ which is true ...
0
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0answers
13 views

$∀ Z^+$ can be written as $c_r * 3^r + c_{r-1} * 3^{r-1} + …. + c_2 * 3^2 + c_1 * 3 + c_0$

I am trying to prove the following statement: $∀ Z^+$ can be written as $c_r * 3^r + c_{r-1} * 3^{r-1} + …. + c_2 * 3^2 + c_1 * 3 + c_0$ where $c_r =$ 1 or 2, and $c_i$ = 0, 1, or 2 for all integers ...
2
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1answer
53 views

Homeomorphism between $\mathbb{R}$ and $\mathbb{Q}$ - why does cardinality matter?

When I look up why $\mathbb{R}$ and $\mathbb{Q}$ are not homeomorphic, almost all the answers just say something along the line of "Because, Cardinality" and then ends there. Can someone provides ...
0
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2answers
25 views

If $G$ is isomorphic to $H$, show ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$

For every $\alpha\in{\rm Aut}(G)$, I've defined $A:H\rightarrow H$ by $$A(h)=\phi(\alpha(\phi^{-1}(h)))$$ where $\phi$ is an isomorphism from G onto H, and I've shown that $A\in {\rm Aut}(H)$. What I ...
0
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2answers
581 views

Use polar complex numbers to find multiplicative inverse

Use the polar form of complex numbers to show that every complex number $z\neq0$ has multiplicative inverse $z^{-1}$. If $z=a+bi$, then the polar form is $z=r(cos(\alpha))+i(sin(\alpha))$. I can do ...
3
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1answer
44 views

What Sort of Discovery Warrants Writing a Paper

I am a high school student who is deeply passionate about mathematics and I have written many different mathematical proofs. I was wondering what sort of discovery warrants writing a mathematical ...
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2answers
14 views

Each regional of a maximal planar graph is a triangle

Show that every region of a maximal planar graph is a triangle. A planar graph G is called maximal planar if the addition of any edge to G creates a nonplanar graph. Will this proof use the ...
0
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1answer
24 views

Why is the following reverse triangle inequality true for given series?

I wish to show that for $(a_k)$ a sequence of numbers, $a_k \in \mathbb{R}$ then claim : $|\sum\limits_{k = n+1}^m a_k | \leq ||\sum\limits_{k = n+1}^\infty a_k| - |\sum\limits_{k = ...
1
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3answers
30 views

{students 1 and 2 are in different groups} vs {students 1, 2, 3, and 4 are in different groups}

Source: Example 1.11, p 26, *Introduction to Probability (1 Ed, 2002) by Bertsekas, Tsitsiklis. Hereafter abbreviate graduate students to GS and undergraduate students to UG. Example 1.11. A ...
2
votes
1answer
32 views

Prove that $|a\sqrt{1-b^2}+b\sqrt{1-a^2}-\sqrt{3(1-a^2)(1-b^2)} +\sqrt{3}ab| \le2$

Prove for any $a, b \in [-1, 1]$ that $$|a\sqrt{1-b^2}+b\sqrt{1-a^2}-\sqrt{3(1-a^2)(1-b^2)} +\sqrt{3}ab| \le2$$ I'm sure there is a solution using the Cauchy-Swartz inequality. Thus i tried to ...
1
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2answers
28 views

Help Proving the Average is greater than B^(1/n)

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Define the following two numbers: $A = (a_1 + a_2 + \cdots + a_n) /n $ (The average of the numbers) $B = (a_1 + a_2 + \cdots + ...
5
votes
3answers
236 views

Prove ten objects can be divided into two groups that balances each other when placed on the two pans of balance. [closed]

There are 10 objects with total weight 20, each of the weight being a positive integer. Given that none of the weights exceed 10, prove ten objects can be divided into two groups that balances each ...
1
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2answers
25 views

Universal Instantiation and Proof of: if $x$ and $y$ are odd integers, then $xy$ is odd.

I have a question about the first part of the proof for the statement "If $x$ and $y$ are odd integers, then $xy$ is odd" if we are using a direct proof. Now i've gotten the proof roughly correct but ...
4
votes
1answer
887 views

If $\lim f(x) = 0,$ then $\lim 1/|f(x)| = \infty.$

The Problem: Suppose that $f:D\to\Bbb R$, where $D$ is a subset of $\Bbb R$ and $a$ is an accumulation point of $D$, $\lim_{x\to a}f(x)=0$, and $f(x)\ne0$ for any $x$ in $D$ in some neighborhood of ...
0
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0answers
10 views

Recovery sequence for semicontinuous functions

I have seen that the next statement holds (if my memory is not wrong) in a certain book. (I forgot which book this is.) For a lower semicontinuous function $f:(0,1)\to\mathbb{R}$ and a given ...
0
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1answer
17 views

Direct proofs involving disjunctions

I've just started a logic and proof class, and I'm confused about what we learned. Given a proof of the form $$(P \lor Q) \rightarrow R$$ why is it true that you only have to show $$P \rightarrow R$$ ...
0
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1answer
21 views

How to prove that the steeple function is not uniformly convergent?

In class we encountered this function $$f_n(x)=\begin{cases} n^2x, & 0 \leq x \leq 1/n\\ 2n - n^2x, & 1/n \leq x \leq 2/n\\ 0, & 2/n \leq x \leq1 \end{cases}$$ The prof said ...
0
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0answers
21 views

Verifying a startegy to prove convexity on partial domain

Assume you have the multivariate function $$f(x_1,x_2,..,x_n)$$ where: $x_i>0 \forall i$, and $\sum_i x_i = 1$. I need to show that $f$ is a convex function. My plan is to show that it is ...
0
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1answer
33 views

Find the limit of $P_{\theta_n}\Big(\sqrt{n}(T_n-\mu(\theta_n))<z_\alpha \sigma(0)-\sqrt{n}(\mu(\theta_n)-\mu(0))\Big)$

Assumptions: Consider a sample of i.i.d random variables $X_i$ $i=1,...,n$, where each $X_i$ is defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X_i:\Omega\rightarrow ...
0
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0answers
35 views

Geometry perpendicular proof

How would I prove that there is a line perpendicular to any given line through a given point not on the line?
3
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3answers
36 views

How to prove these two sets are identical?

This is more a question of the methadology one should use to solve these type of questions: Say there is a set $V \subseteq X \subseteq Y$ and $U \subseteq Y$ such that $$X \setminus V = U \cap X $$ ...
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2answers
327 views

How to show that “Uniformly continuous implies continuous”? [closed]

Assuming that a function f is uniformly continuous, and starting from the ϵ-δ definition of continuity, how does one prove that it is also continuous on the real numbers? Thanks.
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2answers
42 views

If $(d,a)=1$ and $d|ab$ then $d|b$ .

Okay, checking to see if i'm on the right track. I essentially did the same prove for Euclid's lemma but exchanged the $d$ for the $p$. Is that the right idea? Or am I missing something?
2
votes
2answers
126 views

Proof of $\exists x(P(x) \Rightarrow \forall y P(y))$

Exercise 31 of chapter 3.5 in How To Prove It by Velleman is proving this statement: $\exists x(P(x) \Rightarrow \forall y P(y))$. (Note: The proof shouldn't be formal, but in the "usual" ...
1
vote
1answer
100 views

derivative $1 \over x$ -proof

proving $\frac{1}{x}$ by definition $$\left(\frac{1}{x}\right)'=\lim_{h \to 0} {\frac{1}{x+h}-\frac{1}{x}\over h}=\lim _{h \to 0} {\frac{x-x-h}{(x+h)x}\over h}=\lim _{h \to 0} ...
1
vote
1answer
64 views

Prove that $\dim(U+W) + \dim(U\cap W) = \dim U + \dim W$

Let $V$ be a vector space over a field $k$ and let $U$, $W$ be finite-dimensional subspaces of $V$. Prove that $\dim(U+W) + \dim(U\cap W) = \dim U + \dim W$ I'm given that to begin this ...