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
33 views

Long summation question, including sets

I have a really long question I'm absolutely stuck on, I don't even know where to begin: Given: $n \in \mathbb{Z}, \geq 2$ let $S$ be the set of all nonempty subsets of {2,3,...,n}. For each $S_i ...
2
votes
1answer
42 views

Set Operations Question (subtraction, union, intersection)

I have a questions reguarding order of operations for sets: $\forall A,B $ $(A-B) \cup (B - A) \cup (A \cap B) = A \cup B$ If I'm to understand this correctly, the first union $\big((A-B) \cup (B - ...
2
votes
2answers
47 views

Elementary set theory question (not a rational set)

not really sure where to begin with this question: let $$ A = \{x \in \mathbb{R}\space : \cos(x) \in \mathbb{Z}\}$$ and $$B = \{x \in \mathbb{R} : \sin(x) \in \mathbb{Z}\}$$ prove or disprove: ...
0
votes
1answer
26 views

Scalar Products equation proof

$\langle \langle x + y, z \rangle \rangle = \langle \langle x, z \rangle \rangle + \langle \langle y, z \rangle \rangle$ It is clear when there are only $\langle \dot \ , \dot \ \rangle$ but what ...
1
vote
2answers
35 views

How to prove that all of pascal's triangle is composed of integers.

I want to prove that all of n choose k values, i.e. pascal's triangle values are integers. It is pretty obvious, since it is a recursive definition with each term being the sum of its preceding ...
0
votes
1answer
20 views

Let $ L = \infty $ and $ M\neq \infty $ Show that $ \lim_{n \to \infty }(x_n + y_n) = L + M$

$L$ and $M$ are the limits of the sequences $x_n$ and $y_n$ respectively I have already proven for the case where $L,M \in \mathbb{R}$. The method I used doesn't work here where the absolute value ...
0
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0answers
41 views

How can I prove this theorem?

Let n ∈ N. Let b ∈ Z. Then there exists c ∈ Z satisfying c ·n b = 1
2
votes
1answer
63 views

Prove $tr(\mathbf{A} + \mathbf{B} ) = tr(\mathbf{A}) + tr(\mathbf{B})$

Supposedly, this is an easy proof. But I'm really inexperienced and have little mathematical sophistication (trying to improve). Prove $tr(\mathbf{A} + \mathbf{B} ) = tr(\mathbf{A}) + tr(\mathbf{B})$ ...
5
votes
1answer
121 views

Prove spatial velocity identity - screw theory

This question involves a proof regarding coordinate transformations of velocities of screw motions. This comes from "A Mathematical Introduction to Robotic Manipulation" (the text is available for ...
2
votes
3answers
31 views

Cardinality with a Bijection

Suppose that $a, b \in \mathbb{R}: a<b$. Show that $(a, b) ≈ℝ$ by finding a bijection between the sets. I think this might work but am not certain: $g(x) = \frac{2x-b-a}{b-a}$ I was also told ...
1
vote
3answers
48 views

If $f: A→B$ and $g: B→C$ are surjective, then $g\circ f$ is surjective.

In my homework, I wrote: Assume f and g are surjective. Let m be an element of C. then there exists a b that's an element of B, such that g(b) = m and an a element of A such that f(a) = b by ...
0
votes
1answer
21 views

Mapping property of complex fraction field

I recently came across a proof which said that: Suppose $\phi: \mathbb{C}[x]\rightarrow \mathcal{F}$ where $\mathcal{F}$ is a field is a homomorphism. If $ker\phi=0$ then $\phi$ maps isomorphically to ...
0
votes
1answer
18 views

Explaining a. Q(R,T) has unity even if R does not and b. In Q(R,T) every nonzero element of T is a unit

So I've been having trouble understanding this proof for quite a while now. I understand how the field of quotients is formed but not so much of why my professor's answers use this tactic for this ...
3
votes
2answers
62 views

Existence of a normal subgroup in G

Today on my algebra test I had such an exercise: Let $|G|=66$. Show that there is a normal subgroup in $G$ of order $3$. I am not even sure that's true. I wanted to show that $n_{3}$=1. But from ...
0
votes
2answers
55 views

Prove that a set infinite if it has infinite proper subset

Suppose that $A$ is an infinite set and $A \subsetneq C$. Use the definition of "infinite set" to prove that $C$ is infinite also. I am trying to prove that $C$ is infinte. Definition (Infinite ...
0
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3answers
78 views

Prove the derivative of $\sin(1/x)$ exists

How do I prove the derivative of $$\sin(1/x)=-\frac{1}{x^2}\cos(1/x)$$? I understand that you use $$f'(x_0) = \lim_{x \to x_0} \frac{\sin(1/x) - \sin(1/x_0)}{x-x_0} = -\frac{1}{x_0^2}\cos(1/x_0)$$ ...
1
vote
4answers
129 views

Prove that the Fibonacci recursion diverges

I have this sequence with $ n \in \mathbb{N} $ $ f(1) = f(2) = 1 $ and $ f(n) = f(n-1) + f(n-2) $ for $n \ge 3$ I think this sequence is bounded below and unbounded above. So it's clear that this ...
8
votes
1answer
202 views

question on translation of operator proof

Has anyone studied the book 'Nonlinear Partial differential equations with applications' by Tomas Roubicek? I am interested in discussing a point of interest in this book. Specifically, on page 52, ...
1
vote
2answers
32 views

Manipulation of combinations

Let $k,n\in\Bbb N_0$, with $k\le n$. Prove that $$\binom{n+1}{k+1}=\sum_{j=0}^{n-k}\binom{n-j}k\;.$$ Just was hoping someone could give me a hint or two with this problem. I think it has to ...
2
votes
4answers
113 views

Establish the convergence and find the limits of the following sequence

$a_n = \left(1+\dfrac{1}{n}\right)^{n+1}$ I know that the answer is supposed to be $e$ but I am unsure how to reach that answer. I am so lost where to even begin with this
1
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0answers
25 views

Prove that $Q_{8}'=\{1,-1\}$. Is my proof correct?

Prove that Prove that $Q_{8}'=\{1,-1\}$. My proof: $Q_{8}'\neq \{1\}$, because $Q_{8}$ is not abelian. $|Q_{8}/\{-1,1\}|=4$. So $Q_{8}/\{-1,1\} \cong \mathbb{Z}_4$ or $Q_{8}/\{-1,1\} \cong ...
1
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1answer
28 views

Showing that G is solvable

Let $|G|=200$. Show that G is solvable. My beginning of the proof: $|G|=200=2^3*5^2$ Let $n_5$ be the number of Sylow p-subgroups. Then $n_5|8$ and $n_5\equiv1 mod5$. And it implies that $n_5=1$. ...
0
votes
0answers
26 views

$\mathbb R$ ~ $[0,1) \cap (\mathbb R\ $\ $\mathbb Q)$

$\mathbb R$ is the set of the real numbers. $\mathbb Q$ the set of the rational numbers. So how I can prove this? $\mathbb R$ ~ $[0,1) \cap (\mathbb R\ $\ $\mathbb Q)$ I am also not sure what means ...
3
votes
1answer
28 views

Proof: Characterize m

Characterize $m$, an integer, such that $m^2≡1 \pmod{5}$. State your characterization as an "if and only if" statement and then prove it. This question is on my study guide for a test that is on ...
1
vote
1answer
37 views

Showing that $A \subseteq B$ for $A=\{6t\mid t \in \mathrm Z\}$ and $B=\{3t\mid t \in \mathrm Z\}$

Let $A=\{6t\mid t \in \mathrm Z\}$, and $B=\{3t\mid t \in \mathrm Z\}$. Then, show $A$ is a subset of $B$ and prove or disprove that $A = B$. I already know that $A \neq B$, for I can pick a ...
0
votes
1answer
25 views

Prove: The relation $R$ on $\mathbb{N}$ is reflexive, symmetric and transitive

Prove: The relation $R$ on $\mathbb{N}$ given by $mRn$ iff there are natural numbers $p$, $q$ with $m^p$ = $n^q$ is reflexive, symmetric and transitive. Proving $R$ is reflexive: Proof. Suppose $m$ ...
2
votes
2answers
40 views

Prove every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.)

Let $A$ be a commutative ring with unity. Prove: Proof every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.) In the question before ...
0
votes
1answer
104 views

Proof by-contradiction that $(A \subseteq B) \implies (A \setminus B = \{ \})$

I'm studying for an exam and I'm having trouble with one of these problems. Use proof-by-contradiction to prove the predicate $$(A \subseteq B) \implies (A \setminus B = \{ \})$$ where A and B ...
0
votes
1answer
45 views

Is this proof about clock hands lining up correct?

Is http://joshuaoldenburg.com/articles/clock-hands-line-up/ a proof? I.e. does it sufficiently prove the times where the clock hands line up? $$ \begin{align} H &= \text{hour (1-12)} \\ M &= ...
1
vote
2answers
59 views

Prove $1 + \frac{1}{4} + \frac{1}{9} + \cdots + \frac{1}{n^2} < 2 - \frac{1}{n}$ by Induction

The Question Prove $1 + \frac{1}{4} + \frac{1}{9} + \cdots + \frac{1}{n^2} < 2 - \frac{1}{n}$ where $n\ge2$ and $n$ is an integer by Induction My Work Basis Step: 1 + $\frac{1}{4} = ...
1
vote
3answers
42 views

Show that $n!<n^n $ where $n>1$ and is a Positive Integer

Basis Case: $2! = 2\times1 = 2$ $2^2 = 4>2$ Inductive Hypothesis: $k!<k^k$ Induction Step: $k!<k^k$ $k!(k+1) < k^k(k+1)$ $(k+1)! < k^{k+1} + k^k$ I'm confused on where to go ...
0
votes
1answer
39 views

Proving the formula for the cardinality of cartesian products.

Consider sets A and B where |A|=m and |B|=n. Prove by induction on n for a given m that |AB| = mn for all m,n ∈ N where AB = cartesian product. My attempt : Base Case - consider when n = 0 so B is ...
0
votes
1answer
18 views

Proving Bijectivity and Finding the inverse of a function.

I am given this problem: Suppose a, b, c, d ∈ R and ad − bc ≠ 0. Define f : R\ {$d\over -c$} → R\ {$a \over c$} by f(x) is $ax+b \over cx+d$. How do I prove that it is injective and surjective? ...
3
votes
2answers
81 views

Prove that $\Bbb N × \Bbb N$ is countable.

I am given this problem: Prove that $\Bbb N × \Bbb N$ is countable by using the function $f(m, n) = 2^m3^n$ and Theorem that says any subset of a countable set is countable. I'm not exactly sure how ...
0
votes
1answer
16 views

Complex numbers property proof. [duplicate]

I eas given this quesstion in one of my Linear Algebra course with the excercises regarding minimal polynomialsm eigenvalues and diagonalizable matrix: Show that for any two numbers $a,b \in ...
0
votes
4answers
60 views

$|(0,1)| = |\mathbb R|$

For this problem in proving that the cardinality of (0,1) is equal to that of the set of real numbers, would I just prove that (0,1) is uncountable, and then use the theorem that the subset of an ...
0
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0answers
28 views

Define f:Z/3Z→Z/3Z by f([a])=[2a+1]

Just finished proving this to be injective, and well-defined. How would you prove it to be surjective? I understand surjective means that every element in the codomain is being used, and thus is the ...
1
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2answers
21 views

Proving function is not onto

Let A represent the set of real numbers other than $-1$. Consider $f: A \to R$ defined by $f(a) =\frac{ 2a}{ a + 1}$ How would I prove this function is not onto?
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3answers
25 views

Confused about limit proofs conceptually

In a question like this: Prove that if $\lim_{x \to a} f(x) = l$ and $\lim_{x \to a} g(x) = m$ then $\lim_{x \to a} \max(f(x), g(x)) = \max(l, m)$ In general, when asked for proofs like this, are ...
0
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0answers
33 views

Proof of Correctness: Recursion inside loop

I am trying to prove the correctness of the algorithm in the research paper. It is at page 17 in the pdf. ...
0
votes
1answer
45 views

Prove The Limit Does Not Exist

So I have a few questions in which I have to prove that the given sequence does not have a limit and I'm not too sure if I'm on the right track and if I am what is the next step that I have to do. Can ...
0
votes
0answers
33 views

Prove the limit exists

So I have a couple of problems in which I have to prove that the given limit exists and I'm not too sure if I'm on the right track and if I am what it is that I have to do next. Can anybody give me ...
0
votes
0answers
44 views

Show that this Hypergeometric Function equal to this gamma function

I have a question related to hypergeometric functions: Show that ...
6
votes
4answers
894 views

Prove that there is no number that divides both n and n+1

Statement There is no number $x > 1$ that divides both $n$ and $n+1$. Proof (my attempt) Indirect proof: \begin{align} x\mathbin{\vert} n & \implies n = xt_1 \\ x\mathbin{\vert}(n+1) ...
1
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2answers
31 views

Show every bounded infinite set has a maximum limit point and a minimum limit point.

Show every bounded infinite set has a maximum limit point and a minimum limit point. Here is my thought even if it is not formal Let $S$ be bounded and infinite set. Bolzano–Weierstrass theorem: ...
8
votes
3answers
402 views

Instructive examples of elegant, clear, rigorous, terse, but “non-dull” mathematical prose

On the "About" page of the Mathgen project one can read: "More seriously, I think this project says something about the very small and stylized subset of English used in mathematical writing. ...
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0answers
49 views

Writing mathematical proof

i have just finished Multivariate calculus and so far all the mathematics i have done are those calculations sort of questions. However i began to realize that it is not the proper way to do maths and ...
1
vote
2answers
57 views

Suppose G is a group, p is prime , Then the number of elements of G of order p is multiple of (p-1) [closed]

I need Help . "Suppose $G$ is a group, $p$ is prime , Then the number of elements of G of order $p$ is multiple of $(p-1) $". Give me any advise or note
2
votes
1answer
79 views

Prove that two segments are congruent in the arbelos

Background Info + Problem I teach HS Geometry to middle school age students. I generally like to try to solve problems instead of looking up the answer, but this week a student emailed me a problem ...
-1
votes
2answers
38 views

Big-Oh and limits proof?

Prove or disprove: $2^n$ is in $O(3^n)$. I know I have to use some calculus limit techniques but I can't seem to get anywhere. Steps and an approach would be helpful, especially confirming if this has ...