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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
31 views

Discrete Math. Please Help! [on hold]

I'm unsure how to start this proof. Any help or guidance would be greatly appreciated. Thank you in advance. Set F1 = F2 = 1 and for each n ≥ 3 define Fn = F(n−1) + F(n−2). Prove that Fn ≡ 0 (mod 9) ...
0
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2answers
36 views

Show that $ (p \lor (q \land r)) \land p \iff ( \neg p \lor (q \land r) \implies p) $ is valid.

Show that the following logical expression is universally valid. $$ (p \lor (q \land r)) \land p \iff ( \neg p \lor (q \land r) \implies p) $$ Here's what I tried so far: $$[ \ (p \lor (q \land r)) ...
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0answers
33 views

How to show injection from [0,$\infty$) to [2,9] and from [2,9] to [0,$\infty$)? [on hold]

So basically Cantor-Bernsteins theorem. For [0,$\infty$) to [2,9] I tried $$\frac{9-2}{1+x} +2$$ but wolfram alpha showed that it does not contain 2 in it's range. I see why that is, it is because ...
1
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3answers
42 views

Proving uniform convergence of a sequence of functions.

I was wondering how to prove or disprove the following sequence of functions is uniformly convergent $$\ f(n) = \frac{nt}{nt+1}, n≥1, t:[0,1] \to R$$ So far I have analyzed the limits at $t=0$ and $...
3
votes
5answers
66 views

Prove $\sin^{-1}(1)\geq \int_0^b1/\sqrt{1-x^2}dx +(1-b)\pi/2$ for $b \in [0,1)$

I'm trying to prove the following inequality: $$\sin^{-1}(1)\geq\int_0^b1/\sqrt{1-x^2}\,dx +(1-b)\pi/2$$ for every $b \in [0,1)$. I'm given $\sin^{-1}(1) = \pi/2$ and $\sin^{-1}(x)$ is strictly ...
0
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3answers
45 views

Proof that $(n+1)m!>(m+1)!−1$ does not hold for $m>n$

I am currently doing a project that involves some work on Liouville's theorem for transcendental numbers and Liouville's constant. I have found a proof that Liouville's constant is transcendental ...
0
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0answers
18 views

Heine theorem without subsequences

I wanted to prove the following without the need of subsequences : let $f$ be a continuous fonction on a closed interval of $\mathbb{R}$ then it is uniformly continuous. I found this proof but can't ...
0
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1answer
33 views

Prove inverse of strictly monotone increasing function is continuous over the range of original function

Let $f:[a,b] \rightarrow \Bbb R$ be a strictly monotone increasing. Then $f$ has an inverse function $g:[c,d]\rightarrow \Bbb R,$ where $[c,d]$ is the range of $f$. I'm trying to prove that $g$ is ...
3
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1answer
33 views

A corrispondence between maximal ideals

Proposition. Let $R$ a commutative ring with identity. Let $M\ne R$ ad ideal, then $$\bigg (M\;\text{is maximal}\bigg)\iff \bigg(\forall r\in R\setminus M, \exists a\in R\;|\;1+ra\in M\bigg)$$ Proof. ...
0
votes
1answer
22 views

Suggestions for making proof flow better

I am relatively new to writing proofs and I would appreciate some criticism to improve my proof writing skills. Here is proof to prove that if $A^2 = 0$, then 0 is the only eigenvalue of A for my ...
0
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2answers
18 views

Proof by induction coins

Question: Tom only have 2 type of coins: coins: 4 cents and 5 cents. Write a proof by induction that every amount n ≥ a can indeed be paid with Tom coins 1) Base Case: Tom can pay $12, $13, $14, $15, ...
2
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2answers
42 views

Prove the definition of the arcsin(s).

I am given $\arcsin: S \rightarrow (-\pi/2,\pi/2) $ is the inverse function of sin(t) (restricted to [$-\pi/2,\pi/2$]). I'm trying to prove that $\arcsin(s)$= $\int_{0}^{s}1/\sqrt{1-x^2}$ . My ...
2
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1answer
23 views

Showing that an injective Darboux function is strictly monotone.

I was hoping someone could tell me how to prove the following problem I was given: Let $f:[a,b]\to\mathbb{R}$ be a function such that for every $y\in[f(a),f(b)]$ there exists $x\in[a,b]$ such that $y=...
1
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4answers
46 views

Proving that the power series for the cosine function is greater than zero, for $x$ in $[0, \pi/2)$.

I'm trying to prove the cosine power series $$\sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!} \;>\;0$$ for all $x \in [0, \pi/2)$. Here, $\pi$ is defined as the smallest positive real such that $...
0
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2answers
25 views

Proof that $\tan(x)\leq{}x-\frac{\pi}{4}+\tan(\frac{\pi}{4})$ using the Mean Value Theorem

We're asked to proof the above inequality $\forall{}x\in(\frac{-\pi}{2},\frac{\pi}{4}]$. Although am a bit confused with the fact that $\tan(\frac{-\pi}{2})$ is undefined. And thus am stuck when ...
3
votes
2answers
33 views

Help proving there is a sequence of rational numbers

I'm trying to prove the following: Let $\Bbb Q$ be the countable set of rational numbers and $\{x_n\}_{n=1}^\infty$ be a sequence such that for every q $\in$ $\Bbb Q$ there is a $n \in \Bbb N$ with $...
1
vote
1answer
34 views

Associative property in a ring

Let $R$ a ring and we define $$\text{seq}R=\{f=(a_0,a_1,\dots,)\;|\;a_i\in R\}.$$ On this set we define the following operation: let $f=(a_n)_{n\ge0}$ and $g=(b_n)_{n\ge0}$ $$fg=(c_0,c_1,\dots,),$$ ...
0
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0answers
53 views

Are there proofs which are proved by contradiction only? [duplicate]

Are there proofs you know which are, so far as you know, proved by contradiction only? (I thought of the proof that $\sqrt{2}$ is irrational but in Wikipedia there is a direct proof for this). I have ...
1
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2answers
35 views

Differentiability and Continuity on an open interval

Let $f:[0,\infty)\to\mathbb{R}$ be defined by: $$\begin{cases}&x\sin(\frac{1}{x}) \, \, &\text{if}\, \, x > 0\\ & 0, &\text{if} \, \, x = 0\end{cases}$$ Show that $f$ is ...
1
vote
4answers
57 views

Can I prove there is no real solution except $x=0, x=1$, without using the function $W(x)$?

Can I prove there is no real solution except $x=0, x=1$, without using the function $W(x)$? And is it possible to do it without using calculus? $$2^x=x+1.$$ Here is my attempts: $2^x>0 \...
0
votes
2answers
43 views

$1 < a$ and $b\ne0$ imply $1<a^b$

$1 < a$ and $b\ne0$ imply $1<a^b$ when $a,b$ are arbitrary nonnegative integers. I've tried to prove it by induction. I've assumed that $b < a$ (Is valid my assumption?) I'm using this ...
7
votes
7answers
87 views

Prove that if $p$ is prime and $p\le$ $n$ then p does not divide $n!+1$.

Prove that if $p$ is prime and $p\le n$ then $p$ does not divide $n!+1$. I know that in this case since $p$ divides $n!$, then it does not divide $n!+1$ but I am not sure how to show this.
1
vote
1answer
78 views

$a < b$ and $c<d$ imply $a+c < b+d$

$a < b$ and $c<d$ imply $a+c < b+d$ when $a,b,c,d$ are arbitrary nonnegative integers. I know that (assuming we include zero) $$\begin{align*} a<b \Leftrightarrow (\exists x\in \mathbb ...
1
vote
2answers
20 views

How to arrive at these conditions for 2x2 SPD matrices?

Claim: For the matrix $M = \left[ {\begin{array}{cc} a & c \\ c & b \\ \end{array} } \right]$ to be symmetric (trivial) and positive definite: $a>0$ and $ab-c^2>0$ where a,b ...
0
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0answers
41 views

how can i prove $\sum_{t=1}^T a_t.n_t > \sum_{t=1}^T a_t.n'_t $?

in partitioning of numbers (ways of writing a positive integer as a sum of positive integers) . suppose that$ P $and $Q$ are two partitions of $N \in \mathbb N$. $n_t$:=the number of times that a ...
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0answers
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Proving information of integral of $1/(1+t^2)$

For this problem, we are supposed to act as thoug we don't know any properties of the trigonometric function. A(X) = integral from 0 to x of $1/(1+t^2)$dt a.) Why is A(x) a well-defined, ...
0
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1answer
36 views

Proof that a finite series expansion of $f(X)$ at $\alpha$ exists iff $Q(X)$ is a power of $(X-\alpha)$, in $f(X)=\frac{P(X)}{Q(X)}$

I'm working through Gouvea's P-adic numbers book, and early on they give the problem Write $f(X)=\frac{P(X)}{Q(X)}$ in lowest terms, so that $P(X)$ and $Q(X)$ have no common zeros. Show that the ...
0
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2answers
39 views

Basic algebra: Prove that an n-degree polynomial is expressible as the product of n binomials.

Using basic algebra, how does one prove that an n-degree polynomial is expressible as the product of $n$ binomials? Here I am allowing for binomials of the form $xb_i+\alpha_i=x0+1.$ This is ...
0
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1answer
33 views

Find a continuous surjection from $\mathbb{R}^2$ to $Y = \{(x, y) : 0 < x \leq 1, y = \sin (1/x)\}$

I'm looking for $f:\mathbb{R}^2\rightarrow Y$, where $f$ is a continuous surjection in the context of the following problem: -Show the subspace $Y = \{(x, y) : 0 < x \leq 1, y = \sin (1/x)\}$ of $\...
0
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2answers
26 views

Let $a$ and $b$ be two integers. Prove that if $ab=4$, then $(a-b)^3-9(a-b)=0$.

Let $a$ and $b$ be two integers. Prove that if $ab=4$, then $(a-b)^3-9(a-b)=0$. I have tried to approach this by proving the contrapositive instead, but I'm not sure if that's the best approach to ...
1
vote
1answer
17 views

Sigma notation for iterating through number of members of a set with constant expression

Say I have a graph G and I want to sum some constant C (like the minimum degree of the graph) for every vertex. Can I use the following notation? $$\sum_{x \in V(G)}C $$ Is this an appropriate way ...
0
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2answers
19 views

Let $ A =\{n\in\mathbb{Z}: 2 | n\}$ and $B=\{n\in\mathbb{Z}: 4 | n\}$. Prove that $n\in (A - B)$ if and only if $n=2k$ for some odd integer k.

Let $A = \{n\in\mathbb{Z}: 2\, |\, n\}$ and B={$n\in\mathbb{Z}: 4 \, | \, n$}. Prove that $n\in(A - B)$ if and only if $n=2k$ for some odd integer $k$. I'm not sure how to prove this 'correctly'. Any ...
0
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2answers
58 views

Multiplication of blockmatrices

For my university studies I was given this statement to prove: $\begin {pmatrix} A & B \\ C & D\end {pmatrix}\begin {pmatrix} W & X \\ Y & Z\end {pmatrix} = \begin {pmatrix} AW + BY ...
0
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1answer
28 views

Proof area of {$(x,y) R^2 | 0 \le y < e^x$ and $0\le x< h$} is $e^h-1$

I'm supposed to prove that the area of {$(x,y) R^2 | 0 \le y < e^x$ and $0\le x< h$} is $e^h-1$ I was going to try to make it a function and calculate it using a Riemanns sum. That led me to ...
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1answer
25 views

How to change from a double to a single integral changing variables?

So I have the following integral: $$I_1 = \iint u(x,y)dR$$ where $u(x,y)= e^{-(x^2 + y^2)}$ and the region $R$ is the rectangle $[-M,M]\times[-M,M]$. I need to prove that $I_1$ equals: $$I_2 = \...
4
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2answers
91 views

Proof by showing equivalence of derivatives

It can be shown using trig identities that $\cos(2\theta) = \cos^2\theta-\sin^2\theta$. But if we let $f(x) = \sin(2x)$, we can differentiate two ways: 1) $$f(x) = \sin(2x) \rightarrow f(x) = 2\sin(...
0
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4answers
87 views

Prove the following: Let $n$ be an integer. If $2|n^2$, then $4|n^2$.

I came up with the following: $2|n^2$ implies that $2|n*n$. We proved in class that if $q|b*p$, then $q|b$ or $q|p$. Therefore, if $2|n^2$, then $2|n$ or $2|n$. So, $2|n$ implies $n=2k$, for some $k ∈...
1
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0answers
34 views

How would I go about proving that the language accepted by a regular expression is subset of the language accepted by a context free grammar?

Recursive cases: Let A, B be arbitrary RegExps and Let C$_{A}$, C$_{B}$ be cfg(A) and cfg(B) where the properties of cfg can be defined as: cfg($\phi$) = the CFG with no productions cfg($\epsilon$) =...
0
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1answer
25 views

Proof verification of the language of all palindromes as being context-free

Consider that the language L of all palindromes over $\Sigma = \{0,1\}^*$ is not context-free. The following is my attempt at a proof by contradiction. I am new to proof writing and I am wondering ...
0
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2answers
21 views

Proving equality of 3 sets

$A, B, C$ are sets. I have to prove equality of this: $$A \cap C = B \cap C \land A \cup C = B \cup C ⇒ A = B$$ I did this, but I don't know what to do next and whether I even did the right thing: $...
0
votes
5answers
31 views

Help with a proof by induction on polynomial roots

I am currently studying induction. I understand the how the principle is supposed to work, I can follow and understand a proof by induction quite fine (the ones I've seen anyway), but actually using ...
0
votes
3answers
55 views

Delta Epsilon Proof $\lim_{x\to \infty} \frac{x+1}{x+5} =1$

I am trying to prove the following limit using the delta epsilon definition, $$\lim_{x\to \infty} \frac{x+1}{x+5} =1$$ So I want to prove that $$\forall N>0, \exists \epsilon >0| x >N \...
0
votes
0answers
31 views

Proof of Continuity on an Interval

I'm having trouble proving that the function below is continuous on [1,e] $$ \sum_{n=1}^\infty \frac{1}{n3^n} ln(x) $$ Many answers I've seen already involve derivatives (which I cannot use) or ...
0
votes
2answers
31 views

A proof by contradiction worded example

So I have this question. "There are 101 buttons up to 11 different colors in a box. Show that either there are 11 buttons of the same color in the box or there are 11 buttons all in different colors ...
1
vote
0answers
69 views

Injective/Surjective Of Functions Help

So I've created functions based off questions I have and I have to prove the functions are bijections. $A ∪ B$ and $[n + m]$ where $A$ has $n$ elements, $B$ has $m$ elements, and $A ∩ B = ∅$. ...
2
votes
1answer
55 views

Question. Proof of divides - 2 approaches

Suppose $n$ is an integer. If 3|n then 3|$n^2$. Prove. So I'm wondering if both approaches here are ok. $1^{st}$: 3|n so $n=3a$ a in integers $n^2=9a^2$ $n^2=3(3a^2)$ $2^{nd}$: $n=3a$ (both ...
0
votes
2answers
11 views

proof with inequalities simple question

How do you prove: Suppose $x$ is a real number. if $x^3-x>0$ then $x>-1$ It seems really easy to do the contrapositive here i think but dont now how to word it. So suppose $x \le 1$ then $x^...
0
votes
5answers
26 views

proof on difference of two squares and odd integers

How would you prove that every odd integer is a difference of two squares? I re phrased the problem to make it clearer to me: If $k$ is an odd integer then it ca be expressed in the form $a^2-b^2$ ...
0
votes
2answers
23 views

simple proof algebra question on' or'

If $x^2+5y=y^2+5x$ then $x=y$ or $x+y=5$, where $x$ and $y$ are real numbers . Prove this statement. Can someone help me with this problem or how to approach it? I can get x=y Does this mean i have ...