9 votes

Find a rectangular equation from a parametric equation. Why is my approach wrong?

find the rectangular equation for $x = \dfrac{t+1}{t}$ and $y = \dfrac{t - 1}{t}$ $\;\dots\,$ the correct answer is $x^2 - y^2 = 4$ $x^2 - y^2 = 4\,$ is the correct answer for the parametrization $\,...
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8 votes
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Finding where in Ramsey's theorem one uses the Axiom of choice

Ramsey's theorem for countably infinite graphs (or even just Dedekind-infinite graphs) does not require choice. It's only when we try to formulate it for arbitrary infinite graphs that choice comes ...
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7 votes
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Finding the value of $\lim_{a\to \infty}\int_0^1 a^x x^a \,dx$

First, note that $$ \begin{align} \lim_{a\to\infty}\int_0^1a^xx^a\,\mathrm{d}x &\le\lim_{a\to\infty}\int_0^1ax^a\,\mathrm{d}x\tag{1a}\\ &=\lim_{a\to\infty}\frac{a}{a+1}\tag{1b}\\[3pt] &=1\...
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  • 326k
4 votes
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Find a rectangular equation from a parametric equation. Why is my approach wrong?

Notice that $$x^{2}-y^{2}=\left(\frac{t+1}{t}\right)^{2}-\left(\frac{t-1}{t}\right)^{2}=\frac{4}{t},\quad t\not=0$$ But since solving for $y$ and $x$ we get $$\frac{1}{x-1}=\frac{-1}{y-1}\iff y+x=2$$ ...
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4 votes
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Is this valid to prove that $2 \ \vert \ x^3 \Rightarrow 2 \ \vert \ x$?

This method is not valid. You showed that $2(4k^3)\neq x^3$ for all $k\in\mathbb Z$ correctly, but but to show that $2\nmid x^3$, you need to show that $2n\neq x^3$ for all integers $n$, not just ...
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4 votes

Misapplication of the divergence theorem when calculating a surface integral?

I've figured out my mistake, thanks to @Event Horizon. My impression from the Help Center page is that I shouldn't delete my question, so I'll outline what went wrong: I should've applied the ...
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3 votes
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Does this proof of the binomial expansion (a+b)^2 work?

Here is the modified version of your proof with few skipped steps added and properties mentioned. \begin{align} (x+y)^2 &=(x+y).(x+y) & \\ &=(x(x+y))+(y(x+y))&\text{(using property 4 ...
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3 votes
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Correct my strategy about :$\frac{a^3}{2a^2+b^2}+\frac{b^3}{2b^2+c^2}+\frac{c^3}{2c^2+a^2}\geq\frac{a+b+c}{3}.$

I think your strategy is correct. Also, I expanded the expressions after substitution using Python and compared it to your expansion, and the two lists are equal, so at least you didn't make a typo. ...
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3 votes
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How to prove that $\int_{-\infty}^{+\infty}\frac{1+x}{1+x^2}dx$ doesn't exist

Let $$f(u,v)=\ln\left(\frac{1+v^2}{1+u^2}\right)$$ we will prove that $$\lim_{(-u,v)\to(\infty,\infty)}f(u,v)$$ does not exist. if we take $ (-u,u) $ we find $$\lim_{(-u,u)\to(\infty,\infty)}f(u,u)=0$$...
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3 votes
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Proof that $\inf(A)=-\sup(-A)$

You wrote: For each $a \in A$: $$ x \text{ is a lower bound of } A \iff x \leq a$$ This is wrong because the quantifier "For each $a \in A$" is in the wrong place: it should not apply to the ...
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3 votes
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Are proofs in infinite axiom systems verifiable?

I think you've got a bit confused. You can certainly have infinitely big axiom schemas, and that's what Noah's reply in your linked question affirmed. But e.g. in ZFC (which has infinitely many axioms)...
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3 votes

Misapplication of the divergence theorem when calculating a surface integral?

Just for reinforcement, I thought it would be good to show that your work is correct for $\iint_S \vec F \cdot d\vec S$, and we can show this by working directly: $$\vec r (x, y) = \langle x, y, \...
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2 votes

Does this proof of the binomial expansion (a+b)^2 work?

Basically correct, but here are two remarks: From $(x + y)(x + y)$ to $x(x + y) + y(x + y)$ you need the result $(a + b)\cdot c = a \cdot c + b \cdot c$ which can be proved from 1. 4. but you haven't ...
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2 votes
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(Expectation)Consider the joint density $f(x,y)=c(x−y)e^{−x}, 0 \le y \le x$.

$X$ and $Y$ are jointly distributed, so the total probability over the common support $0 \le Y \le X$ must be $1$. In other words, you cannot equate the constant $c$ to some function of $x$. That ...
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  • 112k
2 votes
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Computing $\pi_1$ of subset $X\subseteq \mathbb R^2$, $X$ union of 3 simp.conn. subspaces w/ simp.conn. pairwise intersection but empty intersection

I think the easiest way to calculate $\pi_1(X)\cong\mathbb Z$ is to construct the universal cover. Consider infinitely many disjoint copies of each $V_i$, denoted $\{V_i(k)\}_{k\in \mathbb Z}$. Then ...
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2 votes
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Proving $F(x)$ is a CDF using the PDF

What you have show in that their is a massive step in $F(x)$ at $x=2$. That $F(2)-\lim\limits_{y\uparrow 2}F(2)=1/3$. The function is not left continuous. This does not mean that it is not a CDF, ...
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2 votes

Is there a mistake in an old question on this site: "Where does the sum of sin(n) formula come from?"

hint Yes, Abel's result seems to be false. $$2S\sin(1)=\sum_{k=1}^n\Big((\cos(k-1)-\cos(k))+(\cos(k)-\cos(k+1))\Bigr)=$$ $$1-\cos(n)+\cos(1)-\cos(n+1)=$$ $$2\sin^2(\frac{n+1}{2})+\cos(1)-\cos(n)=$$ $$...
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2 votes

Combinatorics explanation of Inclusion-Exclusion Principle Exactly-$m$ Properties Formula $E_m=\sum_{j=m}^n(-1)^{j-m}\color{blue}{{j\choose m}}S_j$

Here we follow closely the proof of Exercise 3, chapter 2: Sieve methods from Enumerative Combinatrics, Vol. I by R. P. Stanley. To ease comparison I'll try to use OP's notation where appropriate. ...
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2 votes
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($C^1$ Function)False or true? (justify) a) Let $f: X \to \mathbb R$. If $f'(x) = 0, \forall x \in X$, then $f'$ is constant...

a) is False. Let $$X=(-\infty,0)\cup (0,+\infty)$$ and $ f $ defined at $ X $, by $$(\forall x<0)\;\; f(x)=-1$$ and $$(\forall x>0)\;\;f(x)=1$$ $ f $ is differentiable at $ X $ and $$(\forall x\...
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2 votes
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(convex function) Let $f: [a,b] \to \mathbb R$ twice differentiable, and $f''(x) \ge 0$, $\forall x \in [a,b]$....

Definition. a function $f$ on an interval $(a,b)$ is said to be midpoint convex if $$f\left(\frac{x_1 + x_2} {2}\right) \le \frac{1}{2}[f(x_1) + f(x_2)] \tag{1}$$ for all $x_1, x_2 \in (a,b)$. ...
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2 votes

Solving $\tan ^{-1}(\frac{1-x}{1+x})=\frac{1}{2}\tan ^{-1}(x)$

Apply $tan$ to both sides $ \dfrac{1 - x}{1 + x} = \tan \left( \dfrac{1}{2} \tan^{-1}(x) \right) \\ = \dfrac{ x }{ \sqrt{1 + x^2} + 1 } $ From this, $ (1 - x) (\sqrt{1 + x^2} + 1 ) = x (1 + x) $ $ (1 -...
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2 votes

Solving $\tan ^{-1}(\frac{1-x}{1+x})=\frac{1}{2}\tan ^{-1}(x)$

Thank you to @Robin'sPremiumCoffee for a nice algebraic solution. Now the problem with your original method is that although: $$\frac{1-\tan \theta}{1+\tan \theta} = \tan(\frac \pi 4 -\theta)$$ It is ...
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  • 861
2 votes
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Prove that is is not possible to define the connective $\land$ in terms of $\lnot$ and ↔

First thing you should notice is that you also get $\top$ and $\bot$ since $a\leftrightarrow a =\top$. Also notice that $(\neg a\leftrightarrow b)\equiv \neg(a\leftrightarrow b)$. We also can restrict ...
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2 votes
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Proving that $\ast\ast\alpha=(-1)^{kn+k}\alpha$ for every $k$-form $\alpha$ on $\mathbb{R}^n$.

Seems like it's on the right track. But one way to streamline the presentation, and avoid the case work, is to write the sign as $\epsilon_{I,I^c}=\text{sgn}(I,I^c)$, the sign of the permutation $(1,\...
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  • 34.1k
2 votes
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Prove that $\{x\in A:P(x)\}$ is always a subset of $A$

It's by definition, no proof required. Let $x \in P$ be arbitrary, $x\in P \iff x\in A: P(x)$. Hence, $P \subseteq A$. "Technically" there is a conjunction where i put the colon, but we can ...
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2 votes
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Find $\frac{b}{a}$ of trigonometry system of equation.

This problem does not require any Math. Instead, all that is required is visualization and meta-cheating. The $\cos(a) + \cos(b) = 0$ constraint is equivalent to saying that the angles $a$ and $b$ are ...
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  • 21.6k
1 vote

$L^1$ and a.s. convergence

We have $\omega$-wise $$ 2^{k/2} Z_k \not\to 0 \iff Z_k\not=0\quad \forall k\in\Bbb N \iff X^2 \ge 2^k \quad\forall k\in\Bbb N \iff X^2 = +\infty $$ and so: $$ \Bbb P(2^{k/2}Z_k \not\to 0) = \Bbb P( ...
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  • 4,996
1 vote

Let $x \in (0,1).$ How to show that $x \in \bigcup \left(0, \frac{n}{n+1}\right)$?

In order to show that $x$ is in the union; $$\displaystyle\bigcup_{n=1}^{\infty}{\left(0,\frac{n}{n+1}\right)},$$ you'll have to show that $x$ is in one of the intervals of the form; $$\left(0,\frac{n}...
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1 vote
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How do I show that, if $fp=fq$ then $p=q$, in the context of $f$ being the counit of an adjunction and $p,q$ natural transformations?

I want to suggest an alternative solution to you, since your notation is rather involved. Given the adjunction $F\dashv G$ with its unit $\eta:1\rightarrow GF$ and counit $\varepsilon: FG \rightarrow ...
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  • 1,303
1 vote

How to prove that $\int_{-\infty}^{+\infty}\frac{1+x}{1+x^2}dx$ doesn't exist

Alternatively, by definition $$\int_{-\infty}^{+\infty}\frac{1+x}{1+x^{2}}\, {\rm d}x<+\infty \quad \text{ if and only if}\quad \begin{cases} \int_{-\infty}^{\varepsilon}\frac{1+x}{1+x^{2}}\,{\rm d}...
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