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

0
votes
1answer
30 views

Maximal solution of differential equation

Let $K\subset X$ be a compact set and let $x_0\in K$. Suppose that the maximal solution $x(t)$
0
votes
4answers
65 views

For every $x \in [\frac{\pi}{2},\pi]$, $\sin(x)+\cos(x)\geq 1$. Prove rigorously by contradiction.

For every $x \in [o,\frac{\pi}{2}]$, $\sin(x)+\cos(x)\geq 1$. How do you prove this rigorously by contradiction? I understand you start by assuming that $\sin(x)+ \cos(x)<1$ and prove this is a ...
0
votes
1answer
21 views

Proof of a limit formula

If $h(x) = f(x)/g(x)$ $lim(x->b) f(x) = L$ $lim(x->b) g(x) = M$ Prove that $lim(x->b) h(x) = L/M$ Sorry for the terrible latex. ONLY FORMAL PROOFS! For every $\epsilon > 0$ Since ...
0
votes
2answers
18 views

Proof by contradiction how to show is properly

For every $x \in \left[\pi/2,\pi\right]\,,\ \sin\left(x\right) − \cos\left(x\right) \geq 1$. I have drawn the graph and can clearly see that A is true however how do I prove it correctly.
1
vote
1answer
249 views

Is the set of all invertible $n \times n$ matrices a vector space?

I'm studying Algebra and I'm asked to prove or disprove the statement above. I found that is true, but I'm not sure how to prove it. My problem here is that this statement is too "broad", i.e. I ...
1
vote
1answer
140 views

For every $x\in\mathbb R$ and $\varepsilon$ > 0 , there exist $\,q,q'\in\mathbb Q$, such that $q<x<q'$ and $\left |q-q' \right |< \varepsilon$

I'm asked to prove that for every $\varepsilon$ > 0 , there exists two rational numbers $q$ and $q'$ such that $q<x<q'$ and $\left |q-q' \right |<\varepsilon$ where $x$ is a real number. ...
1
vote
1answer
41 views

Prove: $\{α_1,…,α_n\} ⊨ α$ iff $\{α_1,…,α_{n−1}\} ⊨ (α_n→α)$.

Recently began my second logic course and have been surprised at how very, very different it is from the first one. My main struggle is that we have to prove things all the time, and I've never learnt ...
0
votes
1answer
46 views

Limit Delta-Epsilon proof

Prove $\lim_{x \to a} 2x = 2a$ Using the formal proof, not informal. So we know $2|x - a| < \epsilon$ We need to find some $\delta$ We only need to prove there IS SOME $\delta$ right? Only ...
0
votes
2answers
129 views

what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic?

I am trying to get my head around the Rosser-Turquette axiomatisation of Lukasiewicz n-valued logics, but cannot really follow it. Maybe if somebody can give me the axioms for 3 and 4 valued logic ...
0
votes
1answer
14 views

The number of ways to paint a red tile in a grid.

here's the question: "You have nine tiles arranged into a three by three square mosaic. If you color each tile red or blue with equal probability, what is the probability that there exists a two by ...
3
votes
3answers
263 views

Show that inequality holds

How would you show that the following inequality holds? Could you please write your reasoning by solving this problem too? $a^2 + b^2 + c^2 \ge ab + bc + ca$ for all positive integers a, b, c I ...
1
vote
2answers
22 views

Induction on the number of marbles in a heap.

Here is the problem in full: "A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more ...
0
votes
0answers
29 views

Derivations on the space of triangular matrices

I have started to research matrices and have been asked the following. If $d$ is a a derivation on $T_n(\mathbb R)$ and $d(e_{ij})=0$, with $1\le i \le j \le n$, Show that for every $r \in ...
4
votes
1answer
53 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 \ \ ...
6
votes
0answers
216 views
+100

It is easy to show that $\displaystyle S_m=\sum_{n=1}^\infty \frac{n}{2^n + m}$ converges for any natural$\ m$, but what is its value?

In fact the series would converge even if$\ m$ were not natural, I just wanted to state that it is natural in my case. I have found the partial sum formula of$\ S_0$,$\displaystyle \sum_{n=1}^k ...
1
vote
1answer
40 views

Use Fundamental Theorem of Arithmetic to prove that if $a >1$, $p$ is prime, and $p|a ^n$ for some $n \in \mathbb{N}$, then $p|a$

So, by the FTOA, since $a >1$, then a can be broken down into a product of a prime factors, so $a = p_1 \times p_2 \times \dotsm \times p_k$. Then, can I say that since $a$ is multiplied by itself ...
0
votes
0answers
31 views

How would one justify the claim that this differential cannot be solved analytically?

The Wikipedia article on the subject of free fall claims that: when the air density cannot be assumed to be constant, such as for objects or skydivers falling from high altitude, the equation of ...
3
votes
2answers
60 views

Prove that a function is both differentiable and continous at a point $x_0$

Suppose $f$ is differentialble on $(a,b)$, except possibly at $x_0 \in (a,b)$ an is continous on $[a,b]$; assume $ \lim\limits_{x\rightarrow x_0}f´(x)$ exists. Prove that $f$ is differentiable at ...
0
votes
2answers
28 views

If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

Let $n\in\mathbb{N}$. So far I have: If the sum of the digits of $n$ is $k$, then $n = 9m + k$, where $m$ element of an integer (not sure why). Now consider $5n-n$. Help?
0
votes
2answers
43 views

Let n ∈ ℕ. If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

I know how to test the divisibility of a number by 9, but only if I am given what n is. How would I set this problem up?
0
votes
1answer
13 views

Proving that the $k$th element of $A \cup B$ is median of (the first $k$ elements of A) $\cup$ (the first $k$ elements of $b$)

By union here, I am referring to a union where duplicates are allowed. Given two sorted arrays, A and B, how do you prove that the $k$th element in the union of A and B is the median of the following ...
0
votes
1answer
24 views

Prove that if a|c and b|c, and a and b are relatively prime, than ab|c

How do I show this? I have an idea of what to do, but the problem overall is a little confusing to me. I can start the problem, but I just do not see how to get to the solution.
4
votes
3answers
35 views

If d is a norm on V, is $\frac{d(x,y)}{1+d(x,y)}$ a norm on V?

Let d be a norm on a vector space V and let $\psi:V \to [0,\infty)$ be a function defined as $\psi(v)=\frac{d(v)}{1+d(v)}$. Is $\psi$ a norm on $V$? It seems that $\psi$ does not satisfy the ...
0
votes
3answers
32 views

The second derivative of $f^{-1}$ and another question. :)

Suppose both $f$ and $f^{-1}$ are twice differentiable functions. Derive a formula for $(f^{-1})''$. My attempt: We have that by the inverse function theorem that: ...
-2
votes
1answer
49 views

Prove $2n+3 \le 2^n$ for all integers $n \ge 4$.

I have already started the problem but I am unsure on how to proceed. Prove $2n+3 \le 2^n$ for all integers $n \ge 4$. Base Case: Choose $n = 4$. $2n + 3 \le 2^n$ $2(4) + 3 \le 2^4$ $8 + 3 \le ...
0
votes
2answers
41 views

Assume that 495 divides the integer 273x49y5 where x,y ∈ {0,1,2…9}. Find x and y.

So, I know that $495 = 5\times 9\times 11$. So then, if that's the case, then the number $\overline{273x49y5}$ must be divisible by $495$ if and only if it is divisible by 5 and 9 and 11. Then, I ...
-1
votes
0answers
25 views

Prove the continuity and differentiability of a function in a point. [duplicate]

This question Is the same the question as this one (that I have posted yesterday at 12 am that I why I disconected from a large period of time) Prove that a function is both differentiable and ...
-1
votes
2answers
35 views

Geometric summation proof, not calculus

I am trying to take the expression $$T=\sum_{k=1}^nkx^k$$ and make it into a "simpler expression." I have an example similar to it where i am finding $$\sum_{k=1}^nx^k$$ where the answer is $$S_0 = ...
1
vote
3answers
69 views

If $x^2$ is divisible by $4$ then $x$ is even?

I am studying discrete mathematics as course and I have to prove this "If $x^2$ is divisible by $4$ then $x$ is even". I am wondering how to prove it using the contrapositive of this ...
1
vote
6answers
216 views

How to prove a sequence does not converge?

I want to show that the sequence $$a_n=\frac{1}{n}+(-1)^n$$ does not converge to a limit. I know that if a sequence $\left( a_n\right)_{n \in \mathbb{N}}$ converges to a limit L, then ...
0
votes
0answers
9 views

Proving relations are orders

The Problem Let P and Q be posets with respect to some order $\sqsubseteq$. Proof that the following relations are indeed orders. If P' is a subset of P, then P' is also a poset with ...
2
votes
2answers
54 views

Proof of a summation of $k^4$

I am trying to prove $$\sum_{k=1}^n k^4$$ I am supposed to use the method where $$(n+1)^5 = \sum_{k=1}^n(k+1)^5 - \sum_{k=1}^nk^5$$ So I have done that and and after reindexing and a little algebra, ...
1
vote
0answers
100 views

Prove $\sum_{n=1}^\infty \sum_{m=2^{n+1}} ^\infty \frac{(-1)^mn}{m}$ converges and evaluate it.

Noting that the series inside is a part of the alternating harmonic series multiplied by $-n$, we get $\displaystyle \sum_{m=2^{n+1}} ^\infty \frac{(-1)^mn}{m}=\sum_{m=1} ^\infty \frac{(-1)^mn}{m} ...
1
vote
2answers
28 views

Is this a valid method of proof?

We are given that $y = a + b$, and we want to prove that $y = a + c$ (using all the usual properties of numbers that we know from grade school). Does it suffice to set $a + b = a + c$, and by ...
0
votes
2answers
16 views

Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.

I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain. let $a,b\in R$ s.t. $ab\in P$ , ...
0
votes
2answers
69 views

How to determine the domain of $\ln(\sqrt{x^2-3x+2} - x)$?

I know that $$f:x\rightarrow ln(x)$$ is defined $$\forall x>\mathbb{R^{+*}}$$ But what happened when the argument of f is a function as "complicated" as$$\sqrt{x^2-3x+2} - x$$ Obviously we want ...
1
vote
0answers
27 views

Prove that $f$ is uniformly continous

I have to prove this: Suppose $f:(a,b)\to \mathbb{R}$ is differentiable and $|f´(x)| \leq M$ for all $x\in (a,b)$.Prove that $f$ is uniformly continous on $(a,b)$.Give an example of a function ...
0
votes
1answer
17 views

Use the mean value to prove a certain result

I need to prove the following: Use the Mean-Value Theorem to prove that: $$\sqrt{1+h}<1+\frac{1}{2}h$$ for $h>0$ My attempt: we first note that given that $h>0$ then $$1+\frac{1}{2}h ...
1
vote
2answers
32 views

How to formalize proofs

I'm struggling a bit with my discrete maths course and I was wondering if anyone could help me with my assignment. The question I'm working on is, Prove that if a and b are positive integers, then ...
1
vote
1answer
34 views

Proving that $\lim_{ x\to 2} (x^2 - x -3) = -1$ using epsilon and delta

Can someone check if my epsilon delta proof is correct? Thanks a lot!
2
votes
1answer
49 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
4answers
70 views

which one of these proofs more acceptable for $A⊆B$ iff $A\cap B=A$?

I've got two answers to prove that $A⊆B$ iff $A\cap B=A$ first, (*) Assume that $A\cap B=A$. If $x\in A$ and $x\in B$ $\rightarrow $ $x\in (A\cap B)$. And if $y\in B$ but $y\notin A$ $\rightarrow ...
1
vote
2answers
32 views

Proof by induction for $ 2^{k + 1} - 1 > 2k^2 + 2k + 1$ for $k > 4$

I was given this proof for hw. Prove that $ 2^{k + 1} - 1 > 2k^2 + 2k + 1$ So, far I've gotten this Basis: $k = 5$, $2^{5 + 1} - 1 > 2\cdot5^2 + 2\cdot5 + 1$ => $63 > 61$ (So, the basis ...
0
votes
2answers
24 views

Proove the following using either Direct Proof, Contrapostive and Contradiction. (Question related to Geometry).

A circle has centre $(2,4)$. Prove that if $(0,3)$ is not inside the circle, then $(3,1)$ is not inside the circle. I just want to know if my method would be correct. The method I used is as follows: ...
0
votes
1answer
36 views

The limit of product of two functions, where one tends to infinity and the other to zero

If we know that $\lim_{x\to a}f(x)=\infty$, and $\lim_{x\to a}g(x)=0$, then what will be $\lim_{x\to a}(f(x)g(x))$? How do you prove the result? By using examples I can see that the limit can be ...
0
votes
0answers
11 views

Show that $\mathbb{N}=\{ 0\} \sqcup S(n)$ and also that that union is disjoint?

Show that $\mathbb{N}=\{ 0\} \sqcup S(n)$ and also that that union is disjoint? I've been assigned this exercise in my lectures of elements of mathematics 2. Three axioms have been given for a Peano ...
0
votes
2answers
31 views

Let $a$ be an element of order $n$ in a group $G$. If $m$ and $n$ are relatively prime, then $a^m$ has order $n$.

Let $a$ be an element of order $n$ in a group $G$. If $m$ and $n$ are relatively prime, then $a^m$ has order $n$. Assume $m$ and $n$ are relatively prime, and that $a^m$ does not have order ...
-4
votes
0answers
65 views

Geometric Proof for Fermat's Last Theorem - A Question [on hold]

I have been working on a geometric proof for Fermat's last theorem that I just realized has been worked on already in some shape or form (ba-dum-tsh). Before anyone says it, yes, I am aware that this ...
1
vote
1answer
29 views

Let$\ \lim_{n\to \infty} \frac{ \ln n}{f(n)}=1$. If$\ a,b,c$ are natural, can we have$\ a^{b+c \ln n}\sim a^{c f(n)}$?

I shall note that$\ n$ as well goes through the natural numbers and that$\ f(n)$ is rational for any$\ n$. Also, I'm obviously excluding$\ a=1$. I'm inclined to think my claim is not possible ...
2
votes
1answer
110 views

two $\gcd$s that are coprime

Let $a, b$ and $c$ be integers. Prove that if $\gcd(a, b)$ and $\gcd(a, c)$ are coprime, then $\gcd(a, bc)$ = $\gcd(a, b) · \gcd(a, c)$ I am stumped in this problem. Can anybody clarify me what ...