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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0
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1answer
38 views

Solve by induction.

How would I show this equation is odd by using the induction hypothesis: $$ g(s) = 3(g(s-1))+(g(s-2))+1 $$ I was thinking that I would prove $g(s)$ is odd by $g(s+1) = 3(g(s)+g(s-1))+1$. How would I ...
0
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0answers
29 views

trouble in reading statements involving symbols

I have trouble in reading statements involving symbols. (1) For example, when reading a statement from a paragraph: Suppose the size of the set is n. The symbol ...
1
vote
1answer
18 views

Proof by induction for a recursive function f

Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive case: $\operatorname{f}(x) = ...
0
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2answers
36 views

Prove that $ (A \cup B) \cap C \subseteq A \cup (B \cap C)$

I'm trying to practice proof writing, and found the following text question: For all sets A,B,C: $ (A \cup B) \cap C \subseteq A \cup (B \cap C)$ The first step I was thinking of showing is that: ...
2
votes
0answers
42 views

Assuming a Solution Exists

Students who are beginning to learn proofs (and some seasoned pros) occasionally commit the error of assuming what they're trying to prove. My question involves assuming at the onset that a solution ...
1
vote
1answer
22 views

If (an)→ L, an > 0 for all n ∈ N, and L > 0, then prove that √an → √L .

To be honest, I don't even understand what the question is asking, and have no idea how to answer it. Any guidance would be great. I know convergent/divergent definitions, as well as basic limit laws, ...
0
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3answers
30 views

Disproving statement of convergence

How can I disprove this statement: If the sequence $(a_n+b_n)^\infty_{n=1}$ converges, then both $(a_n)^\infty_{n=1}$ and $(b_n)^\infty_{n=1}$ converge. Is this statement able to be disproved? I ...
0
votes
4answers
35 views

Suppose that the function $f: \mathbb {R}\to\mathbb{R}$ is continuous. Define $G(x) = \int_{0}^x (x - t)f(t) dt$, $\forall x$ [closed]

Suppose that the function $f: \mathbb {R}\to\mathbb{R}$ is continuous. Define $G(x) = \int_{0}^x (x - t)f(t) dt$, $\forall x$. Prove $G''(x) = f(x)$ for all $x$.
2
votes
1answer
31 views

Trouble Understanding Proof About Polynomials

In the question I have to prove that: There is no polynomial $P (x) = a_n x^n + a_{n−1}x^{n−1} + · · · + a_0$ with integer coefficients and of degree at least 1 with the property that $P(0), ...
1
vote
3answers
49 views

Prove that the sequence $\cos(n\pi/3)$ does not converge

EDIT: Using a rigorous formal proof, I need to prove that this sequence does not diverge. I of course understand why it doesn't converge... $n=1$ to infinity of course. So, I have a bit of trouble ...
0
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1answer
38 views

Prove the sequence $(\cos(\frac{n\pi}{3}))_{n=1}^\infty$ does not converge

How would I be able to prove that $(\cos(\frac{n\pi}{3}))_{n=1}^\infty$ does not converge? I know that for a sequence to converge to a limit, then for all $\varepsilon > 0, \exists N \in \mathrm N ...
0
votes
1answer
26 views

Proof by induction for a recursive function

Really having a tough time doing this question: Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive ...
1
vote
1answer
55 views

Solve this tautology

Hypotheses: not $q$, $p$ or not $s$, $p \rightarrow$ ($d$ and $q$), $e \rightarrow s$ Conclusion: not $e$ I have thus far, but unsure how to proceed. I am looking forward to solve it using ...
0
votes
2answers
62 views

Prove that $x=0.1234567891011\cdots$ is irrational [duplicate]

Prove that $x=0.1234567891011\cdots$ is irrational Proof: we argue by contradiction.suppose x is rational. then its decimal expansion ultimatetly periodic. Lets p denote the perid of this expansion. ...
0
votes
1answer
33 views

Prove that a function is a linear transformation.

Lets say that I have a vector space $A$ and a linear transformation defined as $f : A → A$. Now I have a function $g : A → A$ defined as $g(a) = bf(a)$ where $a\in A$ and $b \in \mathbb{R}$ is a ...
0
votes
2answers
22 views

Proving an Equality involving Matrices

I have been thinking about this problem for a while and I still can't come up with a solution. Could you please point me in a direction? Here's the problem. ...
1
vote
1answer
12 views

Frequency integration theorem (Laplace transform)

In my textbook I have the following theorem about the integration of the frequency (F(s)): Let the Laplace transform of a function $f(t)$ be $\mathscr{L}\{f(t)\}=F(s)$. If $\dfrac{f(t)}{t}$ is the ...
3
votes
1answer
32 views

Multiplicity of intersection between tangent and elliptic curve

Doubling a point (adding it to itself) on an elliptic curve is done by taking the tangent to the point and calculating the other point where the line intersects the curve. That point is then reflected ...
0
votes
1answer
19 views

Prove that the function is of exponential order and proving in mathematics

I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton: Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that ...
1
vote
2answers
27 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
24 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
43 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
23 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
33 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
19 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
40 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
60 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
110 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
28 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
45 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
19 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
13 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
59 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
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2answers
53 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
73 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
126 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 ...
7
votes
1answer
128 views

Nonlinear partial differential equations with applications

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
31 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
107 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
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vote
0answers
20 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
vote
1answer
26 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
22 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
26 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
36 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
23 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
37 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
97 views

Trouble with by-contradiction proof

I'm studying for an exam and I'm having trouble with one of these problems. ...
0
votes
1answer
43 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
57 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
40 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 ...