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Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

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1answer
21 views

logarithm inequality proof

I'm trying to prove the following Log($A^m$)> Log$(\frac{B^m}{A^m})$ Where $A \in \mathbb{R}$ and $B \in \mathbb{R}$. $m \in [1,\infty]$. Any hints will be appreciated
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0answers
15 views

How to prove that a parametric curve is the intersection of 2 surfaces

So I've been working on a way to prove rigorously that a parametric curve is the intersection of two surfaces but I'm unsure of how to show it Question Show that the curve with parametric equations ...
0
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1answer
41 views

Proving the determinant for an $n\times n$ matrix equals some value

Let $n \in \mathbb{N}$. For every $1 \leq i, j, \leq n$, let $f_{ij}(x)$ be differentiable. Define the $n \times n$ matrix $A(x)$ whose $(i, j)^{\text{th}}$ entry equals $f_{ij}(x)$. Let $F(x) = \...
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2answers
29 views

prove that for any positive real numbers x and y, we have xy^3 <= (1/4)x^4 + (3/4)y^4 [on hold]

I'm having trouble starting this up. i saw somewhere where someones uses (x-y)^4 to start up the proof but i am unsure if it is correct as i got x^4 - 4x^3 y + 6x^2 y^2 - 4xy^3 + y^4.
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1answer
70 views

Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for every $x \in \mathbb{R}$, $f(2x) = 2f(x)$ .

Basically I thought about a kind of modulo 2 equivalence class for real numbers, if that makes sense. With that, and noting that for each number $y \in [1,2)$, the numbers $2y$ and $y/2 $ are not in $ ...
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2answers
55 views

Can you prove that $[(x^2 + 2x + 1)\log(x) – x^2\log(x+1)]/2\log(x)\log(x+1)$ > 1 for all x > 2?

Can you prove that $\frac{(x^2 + 2x + 1)\log(x) – x^2\log(x+1)}{2\log(x)\log(x+1)} > 1$ for all $x > 2$?
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1answer
52 views

Guess and Prove by induction a formula for the $n$-th element in a sequence $b_n$ [on hold]

So I've been given a sequence. The sequence $b_0,b_1,b_2$, ... is defined as follows: $b_0 = 0$, $b_1 = 1/2$, and for integers $n \ge 2$, $b_n = \sqrt{b_{n-1}b_{n-2}} + \frac{3n}{2} - 1.$ My ...
2
votes
1answer
40 views

Prove that any two open intervals are equinumerous.

This is Lay's exercise $8.4.b$. Prove that any two open intervals are equinumerous. Is my proof correct? And even if it is how can I make it better? Is there a better alternative? Consider two ...
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2answers
41 views

Guess and prove Formula for Sequence $b_n$ = $\sqrt{b_{n-1}b_{n-2}} + \frac{3n}{2} - 1.$

Having a really hard time wrapping my head around how to approach this. written out the formula up to $b_5$ but not seeing a pattern at all, I've scoured my notes, google and so on. missing some ...
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0answers
24 views

Help with a proof using the pumping lemma

I am confused with even starting the proof. I understand the pumping lemma: Let A be a language over $\Sigma$. If A is regular, then there exists $p > 0$ (pumping length) such that $∀s∈A$, if $|...
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1answer
50 views

Proving $(ab)^{-1} = a^{-1}b^{-1}$, if $a,b\ne 0$

Having only these axioms: add associativity. add identity. add inverse. add commutative. mul associativity. mul identity. mul inverse. mul commutative. distributive. Prove that $(ab)^{-1} = a^{-1}b^{...
2
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1answer
33 views

Proving a statement using induction

Use the principle of Mathematical Induction to show: $$\sum _ { k = 1 } ^ { n } k x ^ { k - 1 } = \frac { 1 - ( n + 1 ) x ^ { n } + n x ^ { n + 1 } } { ( 1 - x ) ^ { 2 } }$$ for every $n\in\mathbb{N}$ ...
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0answers
14 views

Absolute oscillator in Langton's Ant

We have a simple (or single) block of Langton's Ants colony which includes two ants looking in the same direction. Their positions can be interpreted as knight's walk. The distances between each next ...
28
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7answers
5k views

How to attack “if true, prove it; if not true, give a counterexample” question?

I am taking a basic analysis course. This is a general question that I often encounter in weekly homework. How should we start to attack this type of question: if the statement is true, prove it; if ...
4
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1answer
49 views

Inverse Fermat's theorem

Wiles proved that Fermat's last theorem is true, but... does it stand for inverse case? Does equation $\frac{1}{x^n}+\frac{1}{y^n}=\frac{1}{z^n}$ have no whole number solutions for $n>2$?
1
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1answer
40 views

Let $A,B,X$ be finite sets. Prove that $|A \triangle X| + |X \triangle B| = |A \triangle B| \iff A \cap B \subseteq X \subseteq A \cup B$

I'm trying to do the following exercise: Let $A,B,X$ be finite subsets of a set $U$. Prove that $|A \triangle X| + |X \triangle B| = |A \triangle B| \iff A \cap B \subseteq X \subseteq A \cup B$ ...
0
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1answer
37 views

Prove that $A' \subseteq B$ if and only if $A \cup B = U$

Consider subsets $A$, $B$ and $C$ of the universe $U$ Prove that $A^c \subseteq B$ if and only if $A \cup B = U$ I know that there are two directions for the problem: If $A^c \subseteq B$ then $A ...
1
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0answers
11 views

Surjectivity of composition of functions

Assume $F: X\rightarrow Y$ and $G: Y \rightarrow Z$ are surjective prove that $G \circ F$ is surjective. Let $z\in Z$ Set x$\in X$ to be such that $F(x)=y$ and $G(y)=z$ (these exist because F and G ...
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4answers
59 views

Proving logical equivalences in the statements $(∃𝑥)(𝑃(𝑥) → 𝑄(𝑥))$ and $(∀𝑥)𝑃(𝑥) → (∃𝑥)𝑄(𝑥)$

For this I must show that the two statements $(∃𝑥)(𝑃(𝑥) → 𝑄(𝑥))$ and $(∀𝑥)𝑃(𝑥) → (∃𝑥)𝑄(𝑥)$ are logically equivalent. The issue I'm coming up with is that I'm unsure about the proper methods ...
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vote
2answers
42 views

Does $\neq$ hold true like $=$ holds true under equation manipulation?

In general, when given some equation with an arbitrary number of variables of the form $$f(a,b,c,...)=g(a,b,c,...)$$ we can manipulate the equation and the equality holds true. For example, we can ...
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0answers
36 views

Given $E \subset \mathbb{R^n}$ with $m^*(E) < \infty$. Show that if $E$ is compact then $m^*(E) = \lim_{m \to \infty} m^*(\sigma_m)$

So, here is the formal statement: Let $m^*$ denote the Lebesgue outer measure on $\mathbb{R^n}$, and suppose $E \subset \mathbb{R^n}$ with $m^*(E) < \infty$. Let $\sigma_m = \{x\in \mathbb{R^n} :...
1
vote
1answer
48 views

Fast formula for $\sum_{i=1}^{n} (i \cdot (i!))$, for arbitrary $n \in \mathbb{N}_1$

I've been reading How To Prove It second edition by Daniel J. Velleman, and I've encountered an end-of-subsection exercise I can't answer. On page 286, exercise 10 of subsection 6.3 states: "Find a ...
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3answers
23 views

Prove that finding the maximum element is n-1

How it can be proved that finding the maximum element in the n-element set requires at least n-1 comparisons? I think it requires proof by induction. Thank You in Advance.
0
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1answer
23 views

How to derive the Newton interpolation polynomial from the matrix

Consider the interpolation problem: find the polynomial through a given set of points $(x_0,y_0),...,(x_n,y_n)$. Suppose we want the polynomial in newtonian form: $$N(x)=\sum _{j=0}^k[y_0,\ldots ,y_j]\...
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2answers
25 views

Proof regarding indexed families of sets and intervals

I have the following problem: Let $I$ be the set of real numbers that are greater than $0$. For each $x \in I$, let $A_x$ be the open interval $(0,x)$. Prove that $\cap_{x \in I} A_x = \emptyset$. ...
0
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2answers
45 views

Prove that there is no solution to $4x^3 - y^2 = 1$ where x and y are integers [on hold]

Am i suppose to consider all 4 cases where x and y are odd and even or is there a way around it?
0
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1answer
19 views

How to prove that $ n^b \neq O(n^a) $, if b > a > 1

How can we prove that $n^b \neq O(n^a) $, if b > a > 1 Based on Big-O definition: $n^b \neq O(n^a) \iff |n^b| \le c|n^a|$ I know it's funny but I am stuck here and can't figure it out
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0answers
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Borel $\sigma$-algebra $\mathcal{B}(\overline{\mathbb{R}})$

We consider the family $$\mathcal{I}=\{(a,b)\;|\;-\infty<a<b<+\infty\}\cup\{[-\infty, b)\;|\;b\in\mathbb{R}\}\cup\{(a,+\infty]\;|\;a\in\mathbb{R}\}.$$ I proved that $$\mathcal{B}\big(\...
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2answers
15 views

Prove that given a geometric series, $u_{1+p} \cdot u_{n-p}=u_1 \cdot u_n$

Given a sequence ($u_1; u_2; u_3;...; u_n$), $n\in \mathbb{N}$, of $n$ terms in geometric series, show that for every natural number $0\le p\le n-1$, $$u_{1+p} \cdot u_{n-p}=u_1 \cdot u_n$$ I know ...
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2answers
41 views

Prove the tautology (𝑞⋀(𝑝 → ¬𝑞)) → ¬𝑝

I must prove this tautology using logical equivalences but I can't quite figure it out. I know it has something to do with the fact that not p and p have opposite truth values at all times. Any help ...
0
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2answers
58 views

Let $n$ be an integer. Prove that $2\mid (n^4 - 3)$ if and only if $4 \mid (n^2 + 3)$.

The title says it all. I need help to understand the proof writing in this question.
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0answers
29 views

Proving a statement about the average value of a function

I'm working on a rather complicated problem. And in that problem I (for assumed) that the transient or start-up phenomenon does not influence the average value of a function. My question is, can I ...
0
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1answer
33 views

Let $a, b \in Z$. If a is an even integer and b is an odd integer, then $4 \nmid (a^2 + 2b^2)$.

Any help would be appreciated.I need to prove the statement above.
0
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1answer
19 views

How to prove [𝑝⋀(¬(𝑝 → 𝑞))]⋁(𝑝⋀𝑞) ≡ p

I need to prove this using logical equivalences but I'm stuck near the end. Here is what I've got so far. [𝑝⋀(¬(𝑝 → 𝑞))]⋁(𝑝⋀𝑞) ≡ p [𝑝⋀(¬(𝑝 → 𝑞))] ≡ p - Absorption Laws [𝑝⋀(𝑝 ⋀ ¬𝑞)] ≡ ...
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1answer
19 views

Need help completing a proof (with factorial)

Recently I came across the following identity, but if I try proving it with induction, then I get stuck. $$n! = \sum^n_{k=0}(-1)^{n-k}\binom{n}{k}(k+1)^n$$ While trying my induction step I get the ...
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0answers
50 views

Help with the proof of this symmetric polynomial property

Recently, I found this property of the elementary symmetric polynomial (ESP), further read on ESP: https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial, I tried expressing the ESP in terms ...
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1answer
43 views

Prove that for all integers $a, b, c$ if $a+b^3+c^5=6001$ then at least one of $a,b,c$ is a multiple of three.

Prove that for all integers $a, b, c$ if $a+b^3+c^5=6001$ then at least one of $a,b,c$ is a multiple of three. Do I start with cases? How should I go about proving this? Thanks for your help!
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2answers
29 views

Proving that a sum is even if and only if the stated conditions hold

I am so frustrated about this question and I have been working on it the whole day before I decided to post this question. Let $w,x,y,z$ be nonnegative integers. $$w+x+y=z$$ Prove that $z$ is even ...
0
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1answer
33 views

Prove that if $a\mid(2b+3)$ and $a\mid(3b+5)$, then $a\mid13$. [duplicate]

Prove for all integers $a,b$ that if $a\mid(2b+3)$ and $a\mid(3b+5)$, then $a\mid13$. I am not sure where to start for this question, any help would be greatly appreciated, thanks!
0
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1answer
11 views

Proof of sets $A = ({n \in \mathbb{Z}} | (\exists k \in \mathbb{Z})(n=4k+1)) $, $B = ({n \in \mathbb{Z}} | (\exists k \in \mathbb{Z})(n=4j-7)) $

$A = ({n \in \mathbb{Z}} | (\exists k \in \mathbb{Z})(n=4k+1)) $ $B = ({n \in \mathbb{Z}} | (\exists k \in \mathbb{Z})(n=4j-7)) $ Prove that $A = B$ My attempt: Show that $A \subseteq B$ Let $...
1
vote
2answers
42 views

Verify Proof by Mathematical Induction: $n^2 > 4n+1$

I am just learning proof by mathematical induction and wanted to verify if I got the following proof right Use induction to prove $n^2 > 4n + 1$ Proceed with induction. For $n = 5$. The left ...
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vote
2answers
36 views

Subtracting ratios from each other to find which solution is more concentrated.

3 litres of orange concentrate were mixed with 5 litres of water to make a drink. Later, 2 litres of orange were mixed with 3 litres of water. Which mix is more concentrated? Consider the following ...
15
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5answers
946 views

Do powers of 256 all end by 6 and if so, how to prove it? [duplicate]

I computed the 10 first powers of 256 and I noticed that they all end by 6. ...
-1
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1answer
30 views

True or false: $\left(\mu\left[a,\bar{a}\right]=\mu\left[b,\bar{b}\right]\iff a+\bar{b}=b+\bar{a}\right)\iff\mu$ is a homomorphism?

Note: square brackets $\left[\dots\right]$ are use to indicate parameter lists in function signatures. Addition of ordered pairs in the domain is defined by $$\left<a,\bar{a}\right>+\left<b,...
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2answers
43 views

Prove that any integer amount of currency greater than 17 cents can always be formed. [duplicate]

In a strange country, there are only 4 cent and 7 cent coins. Prove that any integer amount of currency greater than 17 cents can always be formed. Do I use induction to show this? ...
0
votes
3answers
39 views

Prove that there exists a unique set $T$ such that for every set $S$, $S\cup{T}=S$.

Prove that there exists a unique set $T$ such that for every set $S$, $S\cup{T}=S$. So far I have assumed that there exist two sets, $T1$ and $T2$ such that $S\cup{T1}=S$ and $S\cup{T2}=S$. Not sure ...
0
votes
4answers
90 views

Prove gcd(f(n), f(n+1))=1

Let $f: N \implies N$ be the function $f(n)=n^2+n+1$. Prove for all $n \in \mathbb{N}, gcd(f(n), f(n+1))=1$. I was able to prove that both $f(n)$ and $f(n+1)$ are odd for all $n$ but now I am stuck. ...
0
votes
2answers
32 views

Metric spaces and boundedness

Suppose M is a metric space. Assume $A\subseteq M$. $A$ is bounded if $\exists R>0$ such that $\forall x,y \in A: d(x,y) \le R$. I would like to show that the following are equivalent: 1) $A$ is ...
1
vote
1answer
48 views

Deduce $\mathbb Q$ is a dense set $($in real numbers$)$

I am trying to deduce $\mathbb Q$ is a dense set $($in real numbers$)$ i.e. $x, y \in\mathbb Q$, there exists $\alpha$ in $\mathbb Q$ such that $x < \alpha < y$. I have let $x$ and $y$ be real ...
5
votes
1answer
133 views

Derivation for angular acceleration from quaternion profile

Given a profile of unit quaternions $q(t)$ that represents the orientation of a body over time, I like to get the angular acceleration $\dot \omega (t)$. I tried to find a formula myself, but I get ...