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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2
votes
3answers
48 views

Show that $G_{s}$ is a normal subgroup of $G$

Definition: $G_{s}:=\{g \in G: g.s=s\}$ My attempt is the following: We take $g \in G$, and we consider this two sets: $$gG_{s}:=\{gh:h\in G_{s} \}$$ $$G_{s}g:=\{hg :h\in G_{s}\}$$ and we will ...
0
votes
1answer
22 views

Prove that a $\kappa : G/G_{s} \to G.s$ is a bijection

I have to prove that given an action this function $\kappa : G/G_{s} \to G.s$ is a bijection. $$ G/G_{s} \to G.s$$ $$gG_{s} \to g.s$$ Where $G$ is a group and: $G_{s}:=\{g \in G : g.s=s\}$(Isotropy ...
1
vote
1answer
21 views

Finding measure of skewness for binomial distribution

Here's how it was done in my class: $E[(X)_3]= n(n-1)(n-2) p^3$ (Calculated using definition. I understand that part properly.) $E[(X)_2]= n(n-1)p^2$ (Calculated using definition again). Now, ...
4
votes
1answer
58 views

Proving $f(x)$ attains $\max$ or $\min$ when $f(x)\to0$ as $|x|\to\infty$.

Suppose $f:\Bbb R\to\Bbb R$ is continuous such that $f(x)\to0$ as $|x|\to\infty$. Prove that $f$ attains either a maximum or a minimum. My attempt at the question : Given $\epsilon > 0 \ \ ...
1
vote
2answers
91 views

Proof Question using Proof By Contradiction, irrationality of $a + \sqrt[b]{5}$

I answered this question but am not quite sure if i did what was correct. If anything is wrong please point it out, thanks. Question:Prove that any number of the form $a + \sqrt[b]5$ is irrational, ...
0
votes
5answers
46 views

Prove by Induction $F_{2n} = F_{n} * L_{n}$, for n >= 1

Where $F$ is the Fibonacci Sequence, and $L$ is the Lucas Sequence. I need to find the inductive proof of this statement. I've got nearly a page of work in front of me trying to use definitions such ...
0
votes
0answers
35 views

Proof Verification/Alternative to Induction- Well ordering proof

This question grew out of Induction and Maximum Principle, which yours truly asked Sep 23, at 11:51. Due to Mauro Allegranza's suggest, I changed focus so as to first prove the equivalence of $(a)$ ...
1
vote
2answers
59 views

Can this be solve using modular arithmetic? $k$ is prime $\Rightarrow$ $8k+1$ is prime

Is the following statement true or false? $\forall k \in \mathbb{N}, k$ is prime $\Rightarrow$ $8k+1$ is prime The answer is that the statement is false because if $k=7$, then $k$ is prime but ...
-1
votes
2answers
30 views

Use division algorithm and then induction to show 3|(n³+2n) for all ℕ. [duplicate]

For division algorithm, would I do something along the lines of n³+2n = 3q+r and go from there? For induction, I did the base case, which is true, and so then I moved on to the k+1 case, in which I ...
1
vote
1answer
66 views

Prove that if $p\ge 5$ is prime, then $p^2 + 1$ is composite

So, coming off of this question, I know how to find out what the remainder is, so after figuring whether the remainder is $1$ or $5$, would I just plug in $p = 6q + (1\ \text{or}\ 5)$ into $p^2+1$? ...
1
vote
5answers
72 views

Proving by induction $5^{3n} + 2 \cdot 5^{2n} - 5^{n} - 2$ is divisible by $4$

I want to prove the following twice. Once by induction then again by any other method. $$5^{3n} + 2 \cdot 5^{2n} - 5^{n} - 2$$ is a multiple of 4 for all nonnegative integers n. Let n=0 , since it is ...
1
vote
2answers
69 views

Prove that the equation $x^{3}-3x+b=0$ has at most one root in the interval $[-1,1]$

I have to prove that the equation $x^{3}-3x+b=0$ has at most one root in the interval $[-1,1]$. My attempt: We consider the function $g(x)=x^{3}-3x+b$.Now since it is a polynomial it is ...
1
vote
1answer
42 views

Divisibility and Primes

Suppose that $p,q,r$ are prime numbers and $p$ is odd. If $p\,|\,(2q+r)$ and $p\,|\,(2q-r)$, prove that $q=r$. So I'm trying to use the definition of greatest common divisor to come up with two ...
0
votes
3answers
33 views

Find the characteristic of the ring $\mathbb Z_6 \times \mathbb Z_{15}$

My attempt: Let the characteristic be $n$. Then, $n \cdot (1_6, 1_{15}) = (0_6, 0_{15})$, i.e. $n \cdot 1_6=0_6$ and $n \cdot 1_{15}=0_{15}$ The least $n$ for which both are true is $30$, so $30$ ...
3
votes
5answers
91 views

How to select the right modulus to prove that there do not exist integers $a$ and $b$ such that $a^2+b^2=1234567$?

I understand the solution but I don't know how the author decided to start with modulo 4 instead of something else? What is it about the expression $a^2+b2=1234567$ that would trigger us to select ...
0
votes
2answers
31 views

Help me understand the proof of $a \equiv b \mod m \Rightarrow r_m(a)=r_m(b)$

Let: $r_m:\mathbb{Z}\rightarrow R_m$ where $r_m(a)=r\Leftrightarrow a \equiv r \mod m $ and $r\in R_m$ where $R_m$ is the set of residues modulo $m$. I understand the above proof until $r' ...
0
votes
1answer
58 views

Proof involving the Pigeonhole principle

Prove that among any given $n + 1$ positive integers, there are always two whose difference is divisible by $n$ My Answer: Using Pigeonhole principle: From a set of at least $2$ different $n+1$ ...
0
votes
0answers
30 views

What's the correct logical conclusion after proving a value holds?

I had to prove that for every set $s$, the number of subsets with odd cardinalities is $2^{n-1}$. I concluded that this formula holds everytime $|s| \geq 1$ and then I used an inductive process to ...
0
votes
2answers
60 views

Continuity Question in Analysis

Prove that if $$f: A \to \mathbb R$$ is continuous at $a$ and $f(a) > 0$ then there exists $\delta > 0$ such that $$x \in (a - \delta, a + \delta )\cap A \implies f(x) > 0$$ Literally ...
0
votes
1answer
28 views

Linear Algebra: Projection of a Linear Transformation

I am having some confusion of this problem: Let $T:\mathbb{R}^3\to \mathbb{R}^3$. If $T(a,b,c)=(a,b,0)$, show that T is the projection on the xy-plane to the z axis. The following is the ...
1
vote
4answers
77 views

Can anyone explain how to show that $n^{5} -n ≡0$ mod $30$ for every $ n \in \mathbb{N} $

I first tried to answer this using proof by induction, however my problem got more complicated when I got to the induction step. Is there another way of solving this problem?
0
votes
1answer
46 views

If a theorem says “$A \iff B$” and I want to prove $A$, does it suffice to show $A \implies B$?

For example, if there is theorem that says: "$[x] = [y] \iff x \sim y$," and I am asked to prove $[(a,b)] = [(c,d)]$ Is it enough to show that $[(a,b)] = [(c,d)] \implies (a,b) \sim (c,d)$, because ...
0
votes
2answers
52 views

Suppose f : A ---> B and g : B ---> A are functions for which g o f = 1A…

If I were to suppose that $f : A \to B$ and $g : B \to A$ are functions for which $g \circ f = 1_A$, is $f$ always surjective and is $g$ always injective? How would I either prove this or counter it? ...
2
votes
1answer
51 views

Lattice homomorphism

I have an order on $\mathbb R ^ \mathbb R: f \le g $ iff $\forall x \in \mathbb R: f(x) \le g(x)$. Now I have a function $\mathbb R ^ \mathbb R \to \mathbb R ^ \mathbb R: F (f)(x)= f(x)+f(-x)$. ...
1
vote
1answer
69 views

What is the best way to share findings?

I have discovered something. I have some things written in pencil on paper which prove some very important things in number theory. So I want to ask how to write a proof? How do I share my findings? ...
1
vote
1answer
31 views

If $A$ and $B$ are similar, then $A+kI_n$ and $B+kI_n$ are similar for all scalars $k$

Two $n \times n$ matrices $A$ and $B$ are similar, written $A \sim B$, if $B = P^{-1}AP$ for some invertible matrix $P$. How do we prove that $A \sim B$ implies that $A+kI_n \sim B+kI_n$ for all ...
0
votes
2answers
18 views

GCD(m,n) = sm + tn proof

Suppose that m and n are positive integers and that s and t are integers such that gcd(m,n) = sm + tn. Show that s and t cannot both be positive or both be negative. I understand that if both of them ...
1
vote
2answers
34 views

Show that there exists a bijection from $(0,1)$ to $(0,1]$ [duplicate]

I'm having trouble envisioning a bijective relationship that maps $(0,1)$ to $(0,1]$. My professor gave the hint that it can be expressed as a piece-wise function $f(x)$ comprising of two cases: _ ...
2
votes
3answers
37 views

If $V \subseteq W$ and dim$(V)$=dim$(W)$, then $V=W$

If $V,W$ are subspaces of $\mathbb{R^n}$. My proof simply states that if the dimension, $m$, of both subspaces are the same, then we know that $m$ linearly independent vectors in both $V$ and $W$ ...
1
vote
0answers
27 views

bijective function with powerset and symmetric difference

Let $X$ a set and $Y\subseteq X$ and $$f: \mathcal{P}(X)\rightarrow \mathcal{P}(X),\text{ with }A\in \mathcal{P}(X)\mapsto A\,\Delta\, Y$$ ($\Delta:=$ exclusive disjunction) Show that $f$ is ...
2
votes
3answers
292 views

Is this a valid inequality formula?

If $a > b$ and $c > d$ then $ac > bd$ provided $a, b, c, d$ are all positive real ? Thanks!
1
vote
1answer
35 views

Is there a Fitch style system that works with some of the modal logics?

My prof taught us to use trees to prove modal logic arguments. Trees seem to provide a more efficient way to test arguments than Fitch does. However, I find that trees generally, and alethic (modal) ...
2
votes
4answers
266 views

How to deduce the following trig relation?

How can I deduce: $$\sqrt{|x|}\sin(\frac{1}{x}) \le \sqrt{|x|}$$?? I know of the relation. $$\sin(u) \le u$$ $$u = \frac{1}{x}$$ $$\sin(1/x) \le \frac{1}{x}$$ But nothing related to $\sqrt{x}$ ...
1
vote
1answer
34 views

Why are linear combinations of independent standard normal random variables also normally distributed?

My professor has given a list of questions that will not be appearing on my test, with this being one of them. I still feel this is extremely important to understand. How can I prove the following ...
0
votes
1answer
39 views

Let $\alpha$ and $\beta$ be disjoint cycles. Prove for every positive integer n, $(\alpha\beta)^n=\alpha^n\beta^n$

Let $\alpha$ and $\beta$ be disjoint cycles. Say $\alpha = (a_1a_2...a_s)$, $\beta=(b_1b_2...b_r)$. Prove for every positive integer n, $(\alpha\beta)^n=\alpha^n\beta^n$ My proof is as follows: ...
-2
votes
0answers
35 views

Rule or proof for $a^b = b^a$ [duplicate]

For the equation $a^b = b^a$ with the constraint $a \neq b$ I can find two solutions: $a = 2, b = 4$ (or vice versa) and $a = -2, b = -4$ (or vice versa) Is there a rule or a proof to get other ...
3
votes
0answers
49 views

Are there any positive integers $a, b, c, d$ such that both $(a, b, c)$ and $(b, c, d)$ are Pythagorean triples?

Pythagorean triple is a triple of integers $(a, b, c)$ such that $a^2+b^2=c^2$. Is there any Pythagorean triple such that, not only $a^2+b^2$, but also $b^2+c^2$ is a square number? If not, how to ...
8
votes
1answer
573 views

A long nasty limit problem

Does the following limit admit a closed-form? $$\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln 2\pi\right]$$ My professor gives this ...
0
votes
2answers
145 views

Invertible functions - proving that $g \circ f$ is invertible

If I were to suppose that $f : A \to B$ and $g : B \to C$ are functions which are both invertible, how would I go about proving that $g\circ f$ is invertible with $(g \circ f)^{-1} = f^{-1} \circ ...
2
votes
3answers
61 views

Linear Maps: Prove if $T^2 =0$, then $I-T$ is bijective

Let $V$ be a vector space, $T$ is in $L(V)$, Prove: If $T^2 = 0$, then $I - T$ is bijective. the book also gave a hint: in polynomial algebra, $(1-t)(1+t)=(1-t^2)$ I'm not quite sure where to start. ...
0
votes
4answers
42 views

Prove that if P(X) is a subset of P(Y) then X is a subset of Y.

Seems obvious: Prove that if $\mathscr{P}(X)$ is a subset of $\mathscr{P}(Y)$, then $X$ is a subset of $Y$. How to write a formal undeniable proof? Here $\mathscr{P}(X)$ is the power set of ...
1
vote
1answer
28 views

Prove that if X is a subset of Y then X intersect Z is a subset of Y intersect Z for all sets X, Y, Z.

How do you write this proof? Say Y = {a, b, c, d} and X = {a, c} and Z = {a, d, e}. Then X is indeed a subset of Y, however, Z intersect Y is {a, d}, and Z intersect X is just {a}, which is of course ...
0
votes
2answers
34 views

Check my argument that this sequence does not converge.

We want to show that $\{n^2+1\}$ does not converge. It's pretty clear that it doesn't converge, and this is only part of a true/false question so I don't really have to explain it, but I would like to ...
4
votes
4answers
223 views

Mathematical Induction Question, Proof Help [duplicate]

Prove using Mathematical Induction that for all natural numbers ($n>0$): $$ \frac 1 {\sqrt{1}} + \frac 1 {\sqrt{2}} + \cdots + \frac 1 {\sqrt{n}} \ge \sqrt{n}. $$ ...
0
votes
1answer
29 views

Number of ways to order a set of interdependent tasks

I am aware that this question has been asked and answered before here. (Combinatorics/Task Dependency) I'd like some help understanding a part of the answer. Consider the graph shown there: ...
-4
votes
1answer
61 views

What Do We Call a Conjecture When It Is Proved? [closed]

When Legendre's conjecture is proved, what will its new name be? I have researched this and I think law or theorem might be used, but I want to know for sure. Legendre's law of prime numbers between ...
1
vote
1answer
69 views

prove if $b \geq a$, then $a^{b} \geq b^a$

I found that if b = a - 1, then $a^{b} \leq b^{a}$ and if a = b, then $a^{b} = b^{a}$ for obvious reasons. Now, i'm having a hard time figuring out how to prove that if $b \geq a$, then $a^{b} \geq ...
0
votes
3answers
55 views

prove $m^{m-1} < (m-1)^m$ for m > 3

I found that if m > 3 then $m^{m-1} < (m-1)^m$ for m > 3 seems to hold true for a lot of cases. Can someone prove this inductively ?
2
votes
3answers
37 views

Given the sequence $a_0=1, a_1=2, a_2=3, a_n=a_{n-1}+a_{n-2}+a_{n-3}$, prove by strong induction that for $n\geq 0, a_n \leq 2^n$

I've been trying to work this out for some time and I keep getting stuck. Here is what I have thus far: Base Case: $n=0 ; 1 \leq 1$ $n=1 ; 2 \leq 2$ $n=2 ; 3 \leq 4$ Induction hypothesis: ...
1
vote
1answer
51 views

How to write a general proof to prove that for all $m$, $m^n \geq n^m$

After proving $m^n \geq n^m$ for several values of $m$, it can be inferred that for every $m$ there's a $k$ such that if $n \geq k$, $m^n \geq n^m$. In other words, this can be generalized as: For ...