For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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2answers
59 views

Prove equality $a^{\log_b c} = c^{\log_b a}$

I'm try to prove the equality: $$a^{\log_b c} = c^{\log_b a}$$ I'm having trouble finding information regarding this, also I need to figure out why $n^{\log_2 3}$ is better than $3^{\log_2 n}$ as a ...
0
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3answers
81 views

Giving Proof by counterexample

I just started learning college mathematics and one of the things I don't like is giving proofs by counterexamples. My question is how is disproving by giving counterexample is seen by advanced ...
1
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5answers
126 views

How to prove that for all natural numbers, $4^n > n^3$?

This is a problem set I have, it's not a homework but it's very important practice... Send me some hints please, I don't want an answer I need to get it by myself but I'm failing miserably... The ...
1
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3answers
72 views

Proving if $\gcd(c,m)=1$ then $\{x\in \Bbb Z \mid ax\equiv b \pmod m\} =\{x\in \Bbb Z \mid cax\equiv cb \pmod m\}$

Okay so I'm confused on how to approach this question. If $\gcd(c,m)=1$, then $S=T$ where $S=\{x\in \Bbb Z \mid ax\equiv b \pmod m\}$ and $T=\{x\in \Bbb Z \mid cax\equiv cb \pmod m\}$. I know ...
0
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6answers
103 views

Suppose that $x$ and $y$ satisfy $\frac{x}{2} + \frac{y}{3} = 1$. Prove that $x^2 + y^2 > 1$.

Ok , i tried to prove this via Contrapositive setting $x^2 + y^2 \le 1$. After doing some algebra i have arrived at $x \le \sqrt{-y^2}$. I'm fairly sure this isn't right. I also solved for x and y in ...
0
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0answers
29 views

Shorten a proof using Galois connections

Consider a Galois connection: $f:R\rightarrow F$ is a lower adjoint of $r:F\rightarrow R$ for partially ordered sets (actually complete lattices) $F$ and $R$. We have also $f(r(g))=g$ for every $g\in ...
4
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2answers
111 views

$2 \uparrow^n 2 = 4$ and the magnificence of $2$

I was reading up on tetration when I realized: $$2 \uparrow\uparrow 2 = 2\uparrow 2 = 2 \times 2 = 2+2 =4$$ Infact, when generally speaking: $$ 2 \uparrow^n 2 =4$$ Now, I realize that this is because ...
3
votes
3answers
418 views

Prove: Number of Derangement is odd if and only if number of items is even .

let $D_n$ be a number of Derangement of n items . prove that $D_n$ is odd if and only if n is even . i was trying to use induction on the $!n=(n-1)(!(n-1)+!(n-2))$ recurrence relation but i cant ...
6
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8answers
957 views

Trigonometry Identity: Prove that $\sin(a-b)=\sin a \cos b - \cos a \sin b$

First, I do not want a proof using $\sin(a+b)=\sin a \cos b + \cos a \sin b$. Second, I suspect that it has something to do with Euler's formula; $e^{ix}=\cos x + i\sin x$, but I am not sure. Can ...
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1answer
118 views

Modular Arithmetic - Pirate Problem

I was reading an example from my book, and I need further clarification because I don't understand some things. I'm just going to include the $f_1$ part in full detail because $f_2$ and $f_3$ are ...
0
votes
1answer
53 views

Struggling to prove that if $n$ is a non zero integer, and $m > 0 \mid n$ then $m \leq |n|$

i need to prove that if $n$ is a non zero integer, and $m > 0$ and $m \mid n$ ($m$ divides $n$), then $m \le |n|$. I feel like i can do it by a combination of proof by contradiction and cases (ie ...
1
vote
0answers
90 views

So we don't need to choose delta, epsilon, or $N \in \mathbb{N}$ in delta-epsilon or sequence convergence proofs?

(http://math.stackexchange.com/a/700667/85079) I would write the proof with all my bounds $\eta$ and then choose $\eta$ to make the conclusion match the arbitrary $\epsilon$. ...
-1
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1answer
39 views

Prove: $a < a^n$ (more details in description)

Let $\rm\:a\in \mathbb Z.\:$ Prove that if $\rm\: a > 1,$ then for all $\rm\:n > 1, a < a^n.$
0
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2answers
73 views

Prove that there is a real solution of $x=e^{-x}$

I know I have to use the intermediate value theorem but how?
4
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4answers
66 views

Prove that $a \equiv b \pmod{m_1m_2}\implies a \equiv b \pmod {m_1}$

So, I have this problem: if $$a \equiv b \mod(m_1m_2)$$ then (show) $$a \equiv b \mod(m_1)$$. I have to do a proof, but I have no idea where to begin the proof. Can someone help? Proof (Edit): ...
3
votes
2answers
144 views

Geodesic equations and christoffel symbols

I want to learn explicitly proof of the proposition 9.2.3. Which books or lecture notes I can find? Please give me a suggestion. Thank you:)
0
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1answer
72 views

Differential map on the vector space of polynomials: Kernel and Image

Given the $V_n$ is the vector space of polynomials of degree $\leq$ n over $\Bbb R$ So $M_D = \begin{bmatrix} 0 & 1 & ... & 0\\ 0 & 0 & 2 &... 0 \\ 0 & 0 &0 &.\\ . ...
3
votes
3answers
109 views

When does one proof of one direction of an If and Only If proof suffice?

Would someone please explain when this is admissible (please expound on $\color{darkred}{sometimes}$)? In advance of starting an Iff proof, how would one divine/previse if this convenience (of a ...
5
votes
7answers
233 views

Within If and Only If Proofs, why can the proof for one direction be easier than the other?

For $ P \iff Q$, my initial sentiment is that because P and Q are equivalent, the total of two proofs (one for each direction) should entail the equivalent level of "difficulty" or "exertion", as well ...
1
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2answers
39 views

Prove by induction on a string

I want to practice proving the following: Given a binary string s, I want to prove $s$ has the same number of substrings 01 and 10 $\iff$ the first and last character of $s$ is the same. For ...
0
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2answers
23 views

proof by induction - creating summations?

I have two proofs I need to do that I can not figure out how to turn into summations in order to solve. $3|(4^n-1)$ I believe that $|$ is meant to symbolize $3$ divides ... $n!\le n^n$ I have to ...
1
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3answers
34 views

EXCERSICE VERIFICATION: Find where $f(x):=|x|+|x+1|$ is differentiable and calculate its derivative

Could someone verify my excersice? a) $f(x):=|x|+|x+1|$ First, analyse the roots of each absolute value, where they go to zero: $$|x|:=\left\{\begin{matrix} & x& x>0 \\ & x- ...
0
votes
0answers
60 views

Finding the coefficients of a partial differential equation after a change of coordinates.

I'm stuck in one of the mathematical steps of my physical problem. I've been following the derivation of my equations (starting at section 4) from this article Symmetric Euler-Angle Decomposition of ...
3
votes
2answers
120 views

When do we write “we are done”?

This may seem like a bit of a silly question, but I notice that in some proofs (a remarkable amount), the author writes: "We are done." after completing a proof. Is this the equivalent of writing one ...
1
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3answers
49 views

Prove that for all $x$ where $0<x<\pi/2$, $\sin x+\cos x>1$

Prove that for all $x$ where $0<x<\pi/2$, $$\sin x + \cos x > 1.$$ I tried multiple Identities I do not know what I am missing. I have tried changing into different identities.
1
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3answers
162 views

Probability Proof $ P[(A \cap B^c) \cup (A^c \cap B)] = P(A) + P(B) - 2P( A \cap B) $

How would I go about proving this statement: $ P[(A \cap B^c) \cup (A^c \cap B)] = P(A) + P(B) - 2P( A \cap B) $ Describe in English the event where the probability is computed by the expression on ...
0
votes
2answers
28 views

Using the method of induction

Can someone help solve this problem? Prove that if $n≥1$ and $a_1,a_2,….,a_n$ are any real numbers, then $|a_1+a_2+⋯+a_n |≤|a_1 |+|a_2 |+⋯+|a_n |$.
2
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2answers
119 views

Another Epsilon-N Limit Proof Question

How to prove the limit of the following sequence using epsilon-N argument. $$a_n=\frac{3n^2+2n+1}{2n^2+n}$$ I took the limit to be $\frac{3}{2}$ and proceeded with the argument, ...
0
votes
1answer
109 views

Comparison Theorem for Integrals

Problem: Let $a>0$ and $b>a+1$. Use the Comparison Theorem to show that the following integral is convergent: $$\int ^ \infty _0 \frac{x^a}{1+x^b} \ dx$$ My attempt at this was that since ...
0
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3answers
51 views

Help with proof of continuous functions with neighborhood value $N$.

Prove that if $f(x)$ and $g(x)$ are continuous at $c$ and $f(c) < g(c)$ then there is a neighborhood $N$ of $c$ such that $f(x) < g(x)$.
0
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1answer
19 views

Name for proof by logical equivalence

A discussion on ELU stackexchange has led to the question of whether there is a name for the style of proof in which you start with the proposition to be proven and then proceed via a chain of ...
15
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3answers
686 views

Starting sentences with mathematical symbols.

I apologise if this is a duplicate in any way or is too opinion-based. To what extent is it best not to start a sentence with a mathematical symbol? I find that when trying to solve a problem or ...
4
votes
3answers
257 views

Do you need to find the domain of a function to prove injectivity?

I teach math and have had this debate with a fellow teacher. I believe the domain of a function is really not important to determine if a function is or is not injective if there is no "real life" ...
4
votes
3answers
216 views

Proving the Obvious

Is it normal that I have the hardest time when I'm trying to prove statements that are blatantly obvious on a visual and/or intuitive level? For instance, how does one go about formally proving the ...
1
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3answers
52 views

Wrting equations for work rate problems

Consider the following An experienced bricklayer can work twice as fast as an apprentice bricklayer. After the bricklayers work together on a job for 6 h, the experienced bricklayer quits. The ...
1
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0answers
51 views

Proof by induction for all positive integers

$\sum_{j=1}^n j^2 = \frac{n(n+1)(2n + 1)}{6}$ So I did the obvious and plugged one to show that $1^2 = \frac{1(1+1)(2(1) + 1)}{6}$ Now I am trying to show that $1^2 + 2^2 + ... + n^2 + (n + 1)^2 = ...
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0answers
35 views

Understanding 2nd half rank-nullity theorem proof.

I'm trying to understand the second half of the rank-nullity theorem (the part that shows $T(e_{k+1}) \dots T(e_{k+r})$ is independent). Assume $e_1 ,\dots e_k, e_{k+1}, \dots e_{k+r}$,is a basis for ...
0
votes
1answer
41 views

If two matrices are similar, the geometric multiplicities of their eigenvalues are the same

Problem Let $A$ and $B$ be similar matrices. Prove that the geometric multiplicities of the eigenvalues of $A$ and $B$ are the same. [Hint: show that, if $B=P^{-1}AP$, then every eigenvector of ...
2
votes
1answer
138 views

Prove for every two sets $A$ and $B$

Prove for every two sets $A$ and $B$ that $A-B$, ${B-A}$ and $A \cap B$ are pairwise disjoint. I'm really stuck on this one. I know pairwise disjoint means no two elements in $A$ and $B$ are ...
0
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0answers
32 views

Proof sum of permutation

I'm trying to prove: $$P(N) = \sum permutation(A,N)=1 \tag{1}$$ for the particular choice of the set $A = \{ \mu_1, \dots, \mu_n, 1-\mu_1, \dots, 1-\mu_n \}$, where $i = 1, \dots, N$ . So for ...
0
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2answers
69 views

Prove 6 Divides n(n+1)(n+2) [closed]

Let n be an integer such that n >= 1. Prove that 6 divides n(n + 1)(n + 2). Not sure where to start, been stuck for a while
1
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0answers
42 views

Integral Domain and PID Proof

Prove that, in a domain, $(a)=(b)$ iff $a = bu$ for some unit $u$. By $(a)=(b)$, it also means that $a\mid b$ and $b\mid a$ so we can write them as $a=bu$ and $b=av$ for some $u, v \in R$ where $u$ ...
2
votes
1answer
43 views

Divisibility and Principal Ideal Domain Proof

Let R be a ring. Show that a|b iff $b \in (a)$ iff $(b) \subseteq $ (a). I first just want to write out what I know about this statement: a|b means that a divides b or a is divisible by b and there ...
0
votes
3answers
126 views

Maximal Proper Ideal is a field proof

Show that a proper ideal M of a commutative ring R is maximal if and only if R/M is a field. What I know: Because M is a proper ideal $M \neq R$. The ideal M is maximal if it is a maximal element ...
5
votes
1answer
241 views

Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...
2
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4answers
143 views

Proper Ideal and Units Proof

Show that an ideal of I R is proper if and only if it does not contain 1 iff and only if it does not contain any units. (1 is the identity element) I'll need to show 3 parts: (1) $\implies$ (2): ...
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0answers
40 views

Ideal Test Proof

Let $\emptyset \subset I \subseteq R$. Prove that I is an ideal of R if and only if $a-b, ra, ar$, $\in$ $I$ for all $a, b \in I$ and $r \in R$. I know that if I is an ideal in a ring R and $a \in ...
2
votes
1answer
65 views

Integral Domain and no nonzero divisors Proof

Prove that a commutative ring is an integral domain if and only if it has no nonzero zero divisors. I think my main problem is that I'm getting jumbled in the wording! By 'no nonzero zero divisors' ...
1
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1answer
33 views

Need help understanding a specific equality in this proof

Question. Let $f:\mathbb R\to \mathbb R$ be a uniformly continuous function. Show that there exists $a,b>0$ such that $|f(x)|\le a|x|+b,$ $\forall x\in\mathbb R$. Proof. Since $f$ is uniformly ...
0
votes
2answers
145 views

Prove that if M = supS for some nonempty bounded set S, then there exists an increasing sequence sn, of points S such that lim sn = M

Prove that if M = supS for some nonempty bounded set S, then there exists an increasing sequence sn, of points S such that lim sn = M Is this a monotone sequence? Do I need to use Cantor's principle