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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.

0
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1answer
27 views

Proof that any integer $z>1$ can be written as $2x+y$, where $x>y$

Imagine a multiple choice questionnaire with 3 choices $a, b,$ and $c$. At the end the sums of each choice are tallied. It seems it's always possible to have a tie for first, as long as the total ...
0
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1answer
22 views

Prove these differently written de morgan laws

We define $$\overline{\bigcup_{p\in P} Sp} =\bigcap_{p\in P} \overline{Sp}$$ and $$\overline{\bigcap_{p\in P} Sp} =\bigcup_{p\in P} \overline{Sp}$$ Which are just another way to write de morgans laws....
0
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1answer
39 views

Showing that two numbers are the same percent different from their average.

More specifically, consider two real numbers $a,b>0$, and their average $r=\frac{a+b}{2}$. It is the case that $a=r*x$ and $b=r*y$ where $\vert 1-x\vert =\vert 1-y\vert$. For example, let $a=5$ ...
2
votes
1answer
52 views

a < b if and only if a++ ≤ b.

I have to prove that a < b if and only if a++ ≤ b. I am using the book analysis 1 by Terence Tao, which unfortunately has no section for solutions of exercises. both a and b are natural numbers, ...
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2answers
65 views

Proving $2\cosh 2x+ \sinh x = 5$

I have been sitting on this question for quite some time and I haven't been able to prove this identity. Please anybody who can help me here. I am new to hyperbolics. $$2\cosh 2x+ \sinh x = 5$$ I ...
0
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0answers
16 views

Prove infinite markov reward process converges

The following question is obtained from Stanford CS234 Lecture 2 notes, Excercise 3.7 Let $r_i$ denote the reward obtained from transition $s_i\rightarrow s_{i+1}$. Furthermore, the return $G_t$ of a ...
0
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3answers
33 views

Prove that $0< \frac{1}{2^{m}} <y$

If $y$ be a positive real number, show that there exists a natural number $m$ such that $0< \frac{1}{2^{m}} <y$ I think I have to use Archimedean property to prove it. The Archimedean property ...
1
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0answers
22 views

Proving following theorem, Grönwall

Suppose $y:[t_0,+\infty) \to [0,+\infty)$, $t_0 \in \mathbb{R}$, is a non-negative continuous function and $u:[t_0,+\infty)\to\mathbb{R}$ is a non-decreasing continuous function. Let $L\in\mathbb{R}^{+...
0
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1answer
19 views

Show that the set of LES-solutions form a sub-vector space of $\mathbb{R^n}$ exactly when $b_i = 0$

The linear system of equations is given: \begin{align} a_{11}x_1+\dots& +a_{1n}x_n=b_1\\ &\vdots\\ a_{m1}x_1+\dots&+a_{mn}x_n=b_m \end{align} Show that the set of the given linear ...
1
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1answer
33 views

Exact ODE: show that $y$ is a solution iff it is in a level set of $F$

A Differential equation of the form $p(x,y(x))\dot{y}(x)+q(x,y(x))=0\hspace{1cm}(1.1)$ is called exact, if there is a differentiable function $\mathbb{R}^2\mapsto\mathbb{R}$ such that ...
1
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1answer
34 views

Proving $\frac{1+\csc^2A\tan^2C}{1+\csc^2B\tan^2C}=\frac{1+\cot^2A\sin^2C}{1+\cot^2B\sin^2C}$

Prove $$\frac{1+\csc^2A\tan^2C}{1+\csc^2B\tan^2C}=\frac{1+\cot^2A\sin^2C}{1+\cot^2B\sin^2C}$$ I chose to manipulate the left hand side of the equation, by firstly replacing $\cot^2A$ with $\csc^2A-...
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2answers
28 views

Prove that f(x) = absolute value of x when x is not 0 and f(0)=9 on the closed interval [-1, 1] is integrable. [on hold]

I must use the Riemann Integral somehow in this proof, but I am completely lost and have no clue where to begin. Prove that f(x) = absolute value of x when x is not 0 and f(0)=9 on the closed ...
2
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0answers
52 views

Formulating ordinary problems mathematically in order to solve them

I've been thinking about an ordinary problem for which there doesn't seem to exist a solution given its constraints. I was wondering how would one go about formulating the problem mathematically such ...
4
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3answers
108 views

linearly independent solution to second order ODE.

Let $y(t)$ be a nontrivial solution for the second order differential equation $\ddot{x}+a(t)\dot{x}+b(t)x=0$ to determine a solution that is linearly independent from $y$ we set $z(t)=y(t)v(...
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0answers
33 views

Does $n^\prime\ne n^{\prime\prime}$ require proof by contradiction? $n^\prime$ is the successor of $n$.

This is the statement of Peano's axioms I will assume for this discussion: $1$ is a number. To every number $n$ there corresponds exactly one number $n^\prime.$ $n^\prime=m^\prime\implies n=m.$ $n^\...
0
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1answer
36 views

Two Proofs for Open Sets and Metric Subspaces

I have two proofs for the following theorem: Let $(S, d)$ be a metric subspace of $(M, d)$, and let $X$ be a subset of $S$. Then $X$ is open in $S$ if and only if $X = A \cap S$ for some set $A$ ...
1
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2answers
40 views

Proving an inequality involving absolute values

How can I prove the inequality $\left|x\right|+\left|y\right|+\left|z\right|\le\left|x+y-z\right|+\left|y+z-x\right|+\left|z+x-y\right|$ for all $x, y, z$ being real number. Can I prove this by ...
0
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2answers
25 views

Find all pairs $(x, y)$ with $x, y$ real, satisfying the equations: $\sin\frac{(x+y)}2=0$ & $|x| + |y| = 1$

Find all pairs $(x, y)$ with $x, y$ real, satisfying the equations: $\sin\frac{(x+y)}2=0$ & $|x| + |y| = 1$ Working:$\frac{x+y}2=0$ or, $x=-y$ I plotted this. Plotting $|x| + |y| = 1$, I got ...
0
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1answer
26 views

Show that with the exception of (3,5) all twin primes are of the form (6k -1 , 6k +1). [duplicate]

Show that with the exception of $(3,5)$ all twin primes are of the form (6k -1, 6k +1) for some k. My question is: Why we have the number 6 beside k ? Also any hint for the proof is appreciated.
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0answers
19 views

Functions and Relations (MTH 331) [on hold]

Prove: Let $f:A\to B$ be a function, then that $f$ is surjective if and only if for all $Y\subseteq B$, $Y=f(f^{-1}(Y))$.
1
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1answer
18 views

Prove that all terms in an arithmetical equations are equals with border conditions

Would it be possible to prove that there is an equation that includes a number N of unknown numbers that are all equal, between 0 and 1 and whose sum is equal to 1 ? And to find this equation ? I ...
0
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0answers
26 views

Question about how to approach existence in proofs

I am working through some problems in Axler's Linear Algebra Done Right textbook, and I noticed that I haven't really developed an intuitive feel for how to approach existence in the proofs. The idea ...
-2
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1answer
21 views

Let (x+a) be the HCF of $x^2+px+q$ and $x^2+mx+n$. Show that $a=(q-n)/(p-m)$ [closed]

Let $(x+a)$ be the HCF of $x^2+px+q$ and $x^2+mx+n$. Show that $a=(q-n)/(p-m)$.
0
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0answers
26 views

Basic negation of a statement

Statement is $\exists I = (x_o - \frac{1}{n}, x_o + \frac{1}{n}), n \in \mathbb N$, s.t $f(x) > 0 $ $\forall x \in I$ When negating the part after "such that", would it be $f(x) \leq 0$ $\forall ...
2
votes
5answers
85 views

Does the non-commutativity of quaternions follow directly from $\rm i^2=j^2=k^2=ijk=-1$?

All of quaternions are, from what I understand, defined simply by $$\newcommand{\i}{\mathrm{i}} \newcommand{\j}{\mathrm{j}} \newcommand{\k}{\mathrm{k}} \i^2=\j^2=\k^2=\i\j\k=-1$$ It is known that ...
0
votes
1answer
61 views

needing help for proofing $\frac{de^x}{dx}=e^x$

could anybody explain why do we proof $\frac{de^x}{dx}=e^x$ in this way "we know $y=e^x=f(x)$ and $(f(y)^-1)' =\frac{1}{f'(x)}$ ,so $\frac{dy}{dx}=\frac{de^x}{dx}=\frac{dln^-1}{dx}=\frac{df(x)^-1}{dx}...
2
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0answers
30 views

Prove that if $x$ is element of the group G then $H = \lbrace x^n : n \in Z\rbrace$ is a subgroup of $G$.

I am looking to prove Dummit and Foote Chapter $1$ problem $27$. The other proofs I have seen are longer so I feel like there is something I'm missing or my proofs aren't as clear as they should be. ...
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2answers
60 views

If I can rewrite an inequality $f(x,y)> g(x,y)$ as $ h(x,y)>k(x,y)$, does that mean $f(x,y)>g(x,y)$ IFF $h(x,y)>k(x,y)$?

If I can rewrite an inequality $f(x,y)> g(x,y)$ as $ h(x,y)>k(x,y)$, does that mean $f(x,y)>g(x,y)$ If and only if $h(x,y)>k(x,y)$? My reasoning/difficulty: Suppose I can rewrite $f(x,y)&...
0
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3answers
70 views

Prove that a line touches a circle [closed]

Let $I$ be the incenter of $\triangle ABC$. The circle passing through $I$ and centered at $A$ meets the circumcircle of $\triangle ABC$ at points $M$ and $N$. Prove that the line $MN$ touches the ...
0
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0answers
20 views

Proving a sequence does not converge by the definition.

Question:Prove that this sequence does not converge for any $x \in \Bbb R.$ $x_n=(-1)^n(1-\frac{1}{n})$ Definition (Negation):$ \exists \varepsilon \forall N \in \Bbb N \exists n \in \Bbb N, n>N:...
0
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0answers
23 views

Proving a piece-wise function does not converge

Question: Does the function $f(x)=\{1$ if $x\in \Bbb Z $, $0$ if $x \notin \Bbb Z$, tend to zero as $x$ tends to infinity? Not sure how to do the piece-wise function, feel free to edit ...
0
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1answer
17 views

Property of absolute continuity of measures and finite variation

So im trying to prove that if $\mu$ is of finite variation and $\mu <<\nu$, if $|\lambda(E_k)| \rightarrow 0$ then $\mu(E_k) \rightarrow 0$. My attempt was: let's consider the set $E_n = \...
2
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0answers
21 views

Sequence tending to infinity (checking epsilon proof)

Question: Prove that the sequence $(x_n)_{n=1}^\infty$ defined by $x_n=\sqrt[3]{n}+1$ tends to infinity. Definition: $\forall K \in \mathbb{R} \exists N\in \mathbb{N} \forall n \in \mathbb{N} ,n>...
0
votes
1answer
31 views

Red-Black-Tree Insertion & Deletion Complexity proof

I'm struggling with two propositions in my algorithms book. I'm unsure how to proof this. The insertion is abolutely logical that it takes up to O(log(n)) recoloring and at most one restructuring (as ...
3
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1answer
75 views
+50

Exercise about sub-$\sigma$-algebra of $\mathcal{B}(\mathbb{R})$

Let $C=\{(-a, a): a \in \mathbb{R}\}$ and $F=\sigma(C)$. Prove that $F=\mathcal{B}(\mathbb{R})\cap\{A\subseteq\mathbb{R}: A=-A\}$. I don't have problems in proving $F\subseteq \mathcal{B}(\...
0
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1answer
42 views

Prove if $a>1, r,s\in \mathbb{Q}, r>s>0$ , then $a^r>a^s>1$

I have proven that if $a>1$ and $m>n$, then $a^m>a^n>1$ with $m,n \in\mathbb{N}$ But I am having severe problems when I am trying to prove it for $a>1, r,s\in \mathbb{Q}, r>s>0$ ...
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7answers
2k views

Why can't I prove summation identities without guessing?

In order to prove using induction that $$\sum_{k = 1}^n k = \frac{n(n+1)}{2}$$ I have to first guess what the sum is and then show that this guess bears out using induction. This is very unusual ...
0
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1answer
15 views

How to prove $(\mathbb{R}\backslash \mathbb{Q})\cap (x,y)\neq \emptyset$ for $x,y\in \mathbb{R}$ and $x<y$? [duplicate]

How to prove $(\mathbb{R}\backslash \mathbb{Q})\cap (x,y)\neq \emptyset$ for $x,y\in \mathbb{R}$ and $x<y$? Sorry, but I don't even know how to start. Any ideas and impulses?
2
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0answers
45 views

How to prove that $\frac{DE}{CD+CE} =\frac{EF}{BC}=\frac{FD}{AC}$? [closed]

Triangles $\Delta BCD$ and $\Delta ACE$ are erected externally to a triangle $\Delta ABC$ such that $AE = BD$ and $∠BDC +∠AEC = 180^{\circ}$. Let $F$ be a point on the segment $AB$ such that $\frac{AF}...
1
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1answer
59 views

Proof of Yoneda's lemma on wikipedia.

https://en.wikipedia.org/wiki/Yoneda_lemma#Proof The part I'm not getting is the last one: Moreover, any element ${\displaystyle u\in F(A)}$ defines a natural transformation in this way. So we ...
0
votes
1answer
58 views

Prove that: If $a+b$ and $b$ are irrational, then $a+kb$ is irrational.

How can I prove that, If $a+b$ is irrational and $b$ is irrational, then $a+2b$ irrational; where $a,b>0$ More generally, if $a+b$ and $b$ are irrational, then $a+kb$ is irrational; where $...
0
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0answers
46 views

Subgroups of the integers

Want to show: Consider the additive group, $\mathbb{Z}$. Show that the subgroups are of the form $n\mathbb{Z}$, for some $n\in \mathbb{Z}$ Proof: Note that for $1\in \mathbb{Z}$, $\langle 1\...
0
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1answer
20 views

Prove the following claim about skew symmetric matrix $\langle x,Ax\rangle =0$

From https://en.wikipedia.org/wiki/Skew-symmetric_matrix#Vector_space_structure The real $n\times n$ matrix ${\textstyle A}$ is skew-symmetric if and only if ${\displaystyle \langle Ax,y\...
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votes
1answer
47 views

How to prove that $∠PCB = ∠ACD$? [closed]

Let $A$ be a point in the interior of triangle $BCD$ such that $AB·CD = AD·BC$. Point $P$ is symmetrical to the point $A$ with respect to $BD$. Prove that $∠PCB = ∠ACD$. (The problem is from ...
0
votes
3answers
38 views

If we could just use '<' instead of '≤', why are we still using '≤' in many statements?

For example,in " $a\leq b+\epsilon$ if $\forall\epsilon>0$, then $a\leq b$ ",it's impossible to find a case where $a=b+\epsilon$ for very $\epsilon>0$.However,people are still using $\leq$ even ...
1
vote
0answers
35 views

Proofs by counting in 2 ways, infinite sets, etc

I've had some trouble in the past with a few topics such as infinite sets, combinatorics (counting in 2 ways particularly), and bijectivity. Is there any general advice you have to be able to think ...
0
votes
0answers
17 views

Recurrence relation for partition function for pentagonal numbers.

I know the following theorems. Theorem 1 $:$ For $|x|<1$ we have $$\prod\limits_{k=1}^{\infty} \frac {1} {1-x^k} = 1 + \sum\limits_{k=1}^{\infty} p(k)x^k.$$ Theorem 2 $:$ For $|x|<1$ we have $...
1
vote
0answers
25 views

Create & Prove Conjecture - Discrete Math (Proofs)

I am stuck on the following problem: Imagine that a building has been overrun with snakes and rats. To help curb the problem, the building manager decides to offer employees brownie points for ...
0
votes
0answers
28 views

Clarification of the proof of Euler's identity regarding the generating function for partitions.

In reference to this question which I asked here couple of days back but didn't get any answer I am posting this question to clarify whether we can able to extend Euler's identity regarding the ...
1
vote
1answer
25 views

Infinitely many $x_0$s in a sequence, same thing as $\exists N\in \mathbb{N}$ s.t $\forall n \geq N, x_n=x_0$?

Let $\{x_n\}_{n=0}^{\infty}$ be a sequence in $\mathbb{R}$. Is the following implication correct? If so, why? $(x_n)$ contains infinitely many $x_0$s $\iff \exists N\in \mathbb{N}$ s.t $\forall n \...