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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19 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. This ...
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0answers
37 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 ...
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2answers
46 views

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

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
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2answers
24 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 ...
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1answer
27 views

Big-Omega proof using L'Hopital's Rule?

Prove or disprove: $15n^2$ is in $\Omega(3 \times 2^n)$ So we'd have to prove or disprove this statement: $$ \exists c \in\mathbb{R}^+,\,\exists B\in\mathbb{N}, \forall n \in\mathbb{N}, n ≥ B ...
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1answer
10 views

Correctly defined measure

I need to show that the measure is unique (correct definition). Prove that the function $\lambda: \sigma(\mathcal{A}\cap V) \rightarrow [0,1]$ such that $\lambda[(A\cap V)\cup(B\cap V^c)]:=\mu(A)$ ...
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2answers
36 views

If H and K are finite subgroups of G (another proof )

I have a question and it's solution , but I want another proof if there exist . Thanks
2
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2answers
76 views

If $a+1/a$ is an integer, then so is $a^t+1/a^t$ for $t\in\mathbb N$

I need to show if $a$ is in $\mathbb{R}$ but not equal to $0$, and $a+\dfrac{1}{a}$ is integer, $a^t+\dfrac{1}{a^t}$ is also an integer for all $t\in\mathbb N$. Can you provide me some hints please?
2
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2answers
70 views

A positive integer $n$ is such that $1-2x+3x^2-4x^3+5x^4-…-2014x^{2013}+nx^{2014}$ has at least one integer solution. Find $n$.

A positive integer $n$ is such that $$1-2x+3x^2-4x^3+5x^4-...-2014x^{2013}+nx^{2014}$$ has at least one integer solution. Find $n$.
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1answer
33 views

Use Resolution to proove a sentence in First Order Logic

I was just wondering if anyone could tell me if I've solved this problem right. If wrong, I would like to know what I did wrong. "Use resolution to prove Green(Linn) given the information below. You ...
3
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0answers
32 views

Real analysis question: Suprema and Infima

Let $S$ be an ordered set with the $L.U.B$ Property, $S \supset B \neq \varnothing$, $B$ is bounded below. Write $L = \{ l : l \; \text{is a lower bound of } \; B \} $. Then, it follows that ...
0
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1answer
35 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
2
votes
2answers
47 views

How to prove/disprove proof on limits (delta-epsilon)

Prove or disprove: $$ \forall \epsilon > 0, \exists \delta>0, \forall x, y \in \mathbb{R}^+, |x - y| > \delta ⇒ |x + y| > \epsilon $$ I've been trying this for some time now but can't ...
1
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1answer
25 views

big-Oh prove or disprove 2^n is in big-Oh(3^n)

the definiton of Big-Oh says $\exists c\in$R+,$\exists B\in$ N,$\forall n\in$N, $n \geq B$$\implies$$2^n \leq c\times 3^n$. I believe $2^n \in O(3^n)$, but how to prove it? can anyone help. This this ...
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2answers
15 views

Proving a Relation that is a Function by Division Algorithm [duplicate]

Let A=B=$\mathbb{N}$ R is: (a,b)$\in$R iff for some q$\in$Integers a=5q+b WHERE 0$\leq$b<5 Given a relation, show that it's a function. To Show: 1) $\forall$a$\in$A$\exists$b$\in$B((a,b)$\in$R) ...
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1answer
63 views

Contradiction proof for a limit law $f(x) \le g(x)$

Suppose that $f(x) \le g(x)$ for all $x$. Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$, provided these limits exist. I posted a similar question, but this is a different ...
2
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2answers
34 views

Proof by Induction - Wrong common factor

I'm trying to use mathematical induction to prove that $n^3+5n$ is divisible by $6$ for all $n \in \mathbb{Z}^+$. I can only seem to show that it is divisible by $3$, and not by $6$. This is what I ...
0
votes
0answers
7 views

Two affine subspaces parallel

Theorem: Two affine subspaces $V,V'$ of $(X,\overrightarrow{X})$ are said to be parallel $(V\parallel V')$ if there is a translation such that $t_{\vec{u}}(V)=V'$ $V\parallel V' ...
5
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1answer
76 views

Being $N$ a constant $>0$, show $\prod_{p<x}_{p \ \text{prime}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}$.

Related. Show that if $x$ is large enough,$$\prod_{p<x}_{p \ \text{prime}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this paper might be ...
0
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3answers
38 views

Proof by minimum counter example

I need to prove that $n^4-n^2$ is divisible by 12 by minimum counter example. I understand the process but I don't understand how we arrive at m>=7. I have seen different proofs but I still don't know ...
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3answers
70 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
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0answers
11 views

What theorems are used in this following proof of derivatives of log normalizer is moments of sufficient statistics?

The below is the derivation of the proof that shows derivative of log normalizer of exponential family is moments of sufficient statistics \begin{equation} ...
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2answers
17 views

GCD proof using fundamental theorem of arithmetic

prove: $\gcd(m,n)=1$ if and only if $\gcd(m^i,n^r)=1$ I believe you need to do something with fundamental theorem of arithmetic to prove one of the sides. Not quite sure though. Help is appreciated. ...
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1answer
33 views

Combinatorial Argument Proof

Prove: $c(40,5) = c(17,5) + c(17,4) + c(23,1) +...+ c(23,5)$ where c is the binomial coefficient. Can I use a combinatorial argument to prove?
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2answers
42 views

Proof using Induction

Give the induction proof of: $$ 1.2 + 2.3 + k(k+1) = \frac{k(k+1)(k+2)}{3} $$ Is this proof even possible? Not sure how to do.
0
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1answer
13 views

Proof between max independent set cardinal and min vertex cover.

i'm tryign to solve this problem for my graph class, but I don't really know where to start. Be G a graph without isolated vertex,proof that it verifies that $\alpha \leq \beta$, where $\alpha$ is ...
2
votes
1answer
46 views

Finding a proof or a counter example in a programming puzzle

Some years ago I entered a programming contest and this was one of the problems: Binary Granny Summary: Given a positive integer N find 2 positive integers such that $$ x + y = N $$Let X and Y be the ...
0
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1answer
24 views

How to show there are infinite solution to a given Pell's equation?

I was asked to prove the Pell's equation $$x^2-7y^2=1$$ has infinitely many solution. Here is what I did By using Brahmagupta method we can generate infinitely many integer solutions. Is that ...
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2answers
19 views

Define f : Z/3Z → Z/3Z by f ([a]) = [2a + 1].

For this problem, I have to prove the function is well-defined, is surjective, and is injective. For seeing it is well defined, I have this: Assume [a1] = [a2] in the set of equivalence classes Z/3Z. ...
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0answers
20 views

Suppose f: A → B. Define a relation ∼f on A by a ∼f b if f(a) = f(b).

So, as stated above, here is my question: Suppose f: A → B. Define a relation ∼f on A by a ∼f b if f(a) = f(b). First, I have to prove that ~f is an equivalence relation on A. So I need to show that ...
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0answers
20 views

Bijective functions on a finite set

Suppose that A is a finite set and f : A → A and g : A → A are functions. I need to prove that g ◦ f is a bijection if and only if f and g are bijections. So, could I say: Assume g of f is a ...
0
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1answer
23 views

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$.

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$. Prove that $f$ is well-defined, surjective and injective I don't really have a problem with figuring out if it's ...
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2answers
35 views

Finding a counterexample to a function proof

This is my proof: If f and g are surjective, then g ◦ f is surjective, with f: A $\to$ B and g: B $\to$ C. I have successfully proved this, but now I have to disprove the converse by finding a ...
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1answer
20 views

Problems Proving Injectivity and Surjectivity

I have these two functions, in which I have to prove or disprove they are injective and surjective: $f:[0,\infty) \to (0,\infty)$ by $f(x) = \frac{1}{x+1}$. $h:\mathrm R \to \mathrm R$ by $h(x,y) = ...
4
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2answers
51 views

Is $g : \mathbb R →\mathbb R$, $g(x) = |x|$ one-to-one and onto?

So, here is my function, in which I am to prove or disprove both if it is onto and one-to-one: Define $g : \mathbb R →\mathbb R$ by $g(x) = |x|$. For onto, can I say that it is not, because if we ...
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1answer
12 views

Justify each step in the proof sequence

$[A \rightarrow ( B \lor C) ] \land B' \land C' \rightarrow A'$ I know how to read the proof sequence, but I don't know what it means to "justify" each step? Does this mean to just state what each ...
3
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2answers
38 views

GCD Direct Proof

I need to show that if $a,b,c$ are ints such that $\gcd(a,b) = 1$ and $c|(a+b)$, then $\gcd(c,a) = \gcd(c,b) = 1$ I want to try and prove this directly because I think it will be more straightforward ...
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0answers
40 views

Working with the Mobius transformatios and linear algebra.

Let $M=\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) \in GL_{2}(\mathbb{C})$ and we recall that the Möbius transformation attached to $M$ is the map: $z \to ...
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1answer
37 views
0
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1answer
24 views

If $x_k ≥ 0\;\forall \in \mathbb N$, and $y_k$ a bounded sequence, then the series $\sum_{k=1}^\infty x_ky_k$ converges

Hi I'm really struggling with this proof. For a start I'm struggling to believe it's true: For example, if we take $x_k = \dfrac{1}{k^2}$ and $y_k = -k^3$ (which is bounded above by any positive ...
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1answer
32 views

Counting sets and adding an element

Let $A$ be a set with $n$ elements, where $n \in \mathbb{\omega}$. Suppose $s \notin A$, prove that $A \cup \{s\}$ has $n+1$ elements. Here is what I have done so far: By induction, let $P(n):$ if ...
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1answer
53 views

Number theory practice exam questions

Looking for some help for these two practice problems for my exam. I'll explain to you what I have so far and my ideas. So for this problem, I solved part (a) using induction, it wasnt too tricky. ...
0
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0answers
22 views

Proving the limit comparison test

I have the next attempt: Because $0<L< \infty$, we can find two positive and finite numbers, $m$ and $M$, such that $m<L<M$. Now, because $L = lim_{n\to \infty} \frac{a_{n}}{b_{n}}$ we ...
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0answers
42 views

Is my proof right of this result?

Suppose that: $\sum_{n=1}^{\infty}a_{n}$ converges absolutely and $\{b_n\}$ is bounded.Prove that $\sum_{n=1}^{\infty}a_{n}b_n$ converges absolutely. My attempt: Let $M$ be the upper bound of ...
2
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2answers
57 views

Verifying the convergence of a series.

I need to prove that the series $$\sum_{n=0}^{\infty}3^{-n}$$ converges and to find the limit. My attempt: We can express our series as: ...
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1answer
26 views

Prove $|A| \le|C|$ for injection and surjective functions

$A$, $B$ and $C$ are finite sets with $F: A \to B$ a surjection and $G: B \to C$ an injection. Prove $|A| \le |C|$ I could prove it using examples, but not sure how to generally.
2
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2answers
52 views

Proof by Geometric Intuition may Fail!!

In a book on differential forms, I read, "After all, there were times when people took geometric intuition as proof, and later found that their intuition was wrong". I would like to see an ...
1
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1answer
60 views

Induction Proof with Combinations?

Show that for all $n\geq0$ $$\binom{n}{0}3^n+\binom{n}{1}3^{n-1}+\dotsc+ \binom{n}{n-1}3^{1}+\binom{n}{n} $$ $$= \binom{n}{0}5^n-\binom{n}{1}5^{n-1}+\binom{n}{2}5^{n-2}-\binom{n}{3}5^{n-3}+\dotsc ...
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1answer
50 views

Prove $\sin x=3x-2$ has only one real solution

Obviously you can draw a graph, but how would you prove this with calculus?
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3answers
41 views

Combinatorial Argument with Natural Numbers

Give a combinatorial argument to show that all natural numbers n ≥ k ≥ m c(n,k) * c(k,m) = c(n,m) * c((n-m),(k-m)) where c stands for combination.