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
16 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
79 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
15 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
59 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
25 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
vote
2answers
17 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
23 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
31 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
42 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
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0answers
32 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
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0answers
42 views
6
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4answers
891 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
vote
2answers
26 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
379 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. ...
1
vote
0answers
45 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
54 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
71 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
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2answers
36 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 ...
0
votes
1answer
38 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 ...
0
votes
1answer
13 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 $\lambda[(A\cap V)\cup(B\cap V^c)]:=\mu(A)$ ...
0
votes
2answers
44 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
votes
2answers
93 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
votes
2answers
75 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 $n$.
0
votes
1answer
52 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
votes
0answers
37 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
votes
1answer
40 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 ⇒ ...
3
votes
2answers
61 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
vote
1answer
31 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 ...
-2
votes
2answers
24 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) ...
0
votes
1answer
81 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
votes
2answers
35 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
11 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
votes
1answer
84 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
votes
3answers
40 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 ...
7
votes
3answers
83 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 ...
0
votes
0answers
14 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} ...
0
votes
2answers
23 views

GCD proof using fundamental theorem of arithmetic

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.
-2
votes
1answer
36 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?
0
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2answers
47 views

Proof using Induction

Give the induction proof of: $$ k(k+5) = \frac{k}{5} $$ Is this proof even possible? Not sure how to do.
0
votes
1answer
14 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
48 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 ...
1
vote
1answer
26 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 ...
0
votes
2answers
29 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. ...
-1
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0answers
27 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 ...
0
votes
0answers
25 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
votes
1answer
24 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 ...
1
vote
2answers
42 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 ...
1
vote
1answer
26 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
votes
2answers
59 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 ...
0
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2answers
19 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 ...