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.

learn more… | top users | synonyms

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
90 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
28 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
59 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
73 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
291 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
34 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
37 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
568 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
41 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
26 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
219 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 ...
0
votes
1answer
36 views

Big Omega Proof for $5^n = \Omega(6^n)$

I got the Big-O ($\mathcal{O}$) proof but I am having troubles with the Big Omega ($\Omega$) proof. I am trying to prove $6^n$ is not the tightest bound for $5^n$. Is this logic true: $7^n$, $8^n$ ...
0
votes
3answers
66 views

Proof by induction $n^2 \geq n+1 \ \forall n \geq2$

I have to prove by induction that $n^2 \geq n+1 \ \forall n \geq2$. I have done the following reasoning: the base case is easy to verify; supposing that $n^2 \geq n+1 $ is true, we prove $(n+1)^2 ...
1
vote
2answers
33 views

$\mathbb{C}$ Forms a Vector Space Over $\mathbb{R}^2$ Proof Question

In my Mathematical Techniques course we've been talking about vector spaces, bases, etc. There is one problem however that I cannot get my head around and that is to prove that $\mathbb{C}$ can be ...
1
vote
0answers
31 views

Correct process for proof in graph theory.

I'm working on what I'm sure is a fairly basic proof in graph theory. I must prove that $Every$ $graph$ $G$ $contains$ $a$ $path$ $with$ $\delta(G)$ $edges$. $\delta(G)$ is the minimum degree of the ...
0
votes
3answers
51 views

Show that 1 is the supremum of $S = \{ x \epsilon R: x^2 < x \}$

I'm still new to proof writing so I was wondering if I could have a little help organizing my thoughts on this, I attempted this proof in a slightly oblong way that is probably not the standard, but I ...
0
votes
1answer
31 views

rewrite $(\sqrt{5} + 2)(\sqrt{5}-2) = 1 $ until…

"rewrite $(\sqrt{5} + 2)(\sqrt{5}-2) = 1 $ until you have an equation with $\sqrt{5}$ on the left and a ratio of two expressions involving $\sqrt{5}$ on the right." Ok..All i need to know is if i'm ...
0
votes
1answer
25 views

If $a \in \mathbb{N}$, prove that gcd$(a, a+2)$ is $1$ if $a$ is odd and $2$ if $a$ is even.

Once again the problem is: If 'a' is an element of N, prove that gcd(a, a+2) is 1 if 'a' is an odd number, and 2 is 'a' is an even number. I really have no idea on how to prove this, and I'm brand ...
0
votes
1answer
43 views

Proof about a subset of a metric space

Prove that a subset $A$ of metric subspace $(P, p')$ of metric space $(M, p)$ is open in subspace $(P, p')$, regarded as a metric space in its own right, if and only if there exists an open set $U$ in ...
0
votes
1answer
23 views

Introduction chapter Exercise Q3 from “How to Prove It: A Structured Approach”

The following question is from the book "How to Prove It: A Structured Approach" Second Edition. Theorem 3 : There are infinitely many prime numbers. Euclid's proof Introduction Chapter : Exercise ...
0
votes
3answers
51 views

Is my arithmetical proof using induction correct?

The exercise 2.b of my textbook ask me to prove that: $$\text{(P): }\;\forall n\in \mathbb{N}, 13\;|\;(3^{n+2}+4^{2\cdot n+1})$$ I would like to know if my proof is correct and if not what I need to ...
6
votes
4answers
1k views

There exist no integers for which $x^2-4y=2$

I am working on a new exercise in my textbook: $$\text{Prove that: (P): }\;\nexists \;x,y \in \mathbb{Z}, x^2-4\cdot y = 2 $$ I am stuck and I would really like to see a correct proof so I can move ...
2
votes
2answers
31 views

Induction Proof without Explictly Using The Induction Hypothesis?

I have encountered several problems where one can prove the desired result without actually needing the induction hypothesis. More specifically, you basically just pick $n \in \mathbb{N}$ and run ...