João Victor Bateli Romão
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Let $A = \{2,3,4\}$ and $B = \{a,b\}$. List elements of $A\times B$.
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3 votes

The terms of the ordered pairs must be separated by a comma. The right answer is $\{(2,a),(3,a),(4,a),(2,b),(3,b),(4,b)\}$.

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If $P \lor Q$ and $\neg P \lor \neg Q$ are both true, do we get a contradiction?
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2 votes

Let $R \equiv P \lor Q$ and $S \equiv \neg P \lor \neg Q$. When you say "when both $R$ and $S$ are true", you are looking for $R \land S$. If $R \land S$ is always false no matter who is $P$ or $Q$, ...

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why $\int \sqrt{(\sin x)^2}\, \mathrm{d}x = \int |\sin x| \,\mathrm{d}x$
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2 votes

The definition of the square root, $\sqrt{\,\cdot\,}$, in real numbers is a function that given a non-negative number $y$, we take the non-negative number $x$ such that $x^2 = y$. That's why we write $...

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How big can a set be?
2 votes

No, there is not. Not in the standard set theory, which is the ZFC Set Theory. The reason is the power set (the set of all subsets of a given set), as explained by the other answers here, we have $|\...

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Evaluation of convolutions by Fejér kernels have finite-dimentional range?
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1 votes

Ok, finally got it. Define, for every $k\in \{0,\dots, n\}^N$, \begin{align*} r_k: \mathbb{R}^N& \to (\mathrm{Lip}(\mathbb{R}^N))^\ast\\ x & \mapsto \begin{aligned}[t] ...

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Why does existence of directional derivatives not imply differentiability?
1 votes

Hint: A function $f:U\subset\mathbb{R}^m\rightarrow \mathbb{R}^n$ is differentiable at $a \in U$ if there exists a linear transformation $T_a:\mathbb{R}^m\rightarrow \mathbb{R}^n$ such that $$f(a+h) =...

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If $a_n$ is a null sequence, does $\sum^{\infty}_{n=1}a_n$ converge?
1 votes

Since $\lim\limits_{n\rightarrow\infty} a_n = a$, then $\forall\varepsilon>0, \exists n_0\in \mathbb{N}$ such that $n>n_0\Rightarrow |a_n-a|<\varepsilon/2$. For $n>n_0$, we have \begin{...

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What is the value of $\ln(\ln(i))$?
1 votes

Hint: Given $z = r e^{i\theta}$, we have $$\log z = \log r +i\theta,$$ and, of course, $$\log r = \int_1^r \frac{1}{t} \mathrm{d}t.$$

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Limit of a sequence bounded by a convergent sequence
1 votes

$\forall \varepsilon>0, \exists n_0\in \mathbb{N}$ such that $n>n_0\Rightarrow 1/n<\varepsilon$. By assumption, $$|X_n|<\frac{1}{n}<\varepsilon,\quad \forall n>n_0,$$ which is ...

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what is the interval of integration of $\int^a_b \frac {dx}{dt}dt $
1 votes

Using the Change of Variables Theorem: Let $f:[a,b]\rightarrow \mathbb{R}$ a continuous function and $g:[c,d]\rightarrow \mathbb{R}$ with continuous derivative and $g([c,d])\subseteq [a,b]$. Then ...

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How many combinations are there of a list of $2000$ items, used $3$ times?
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1 votes

No. If you want a combination, then "Apple-Orange-Boat" is the same as "Orange-Boat-Apple". The order does not matter and therefore $2000^3$ is counting too much. For combination with repetitions, ...

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Mazur Lemma for sequence of Lipschitz functions
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0 votes

Using a diagonal argument, we can consider $(L_n)_{n\in\mathbb{N}}$ such that $\|L_n\|_{\mathrm{Lip}} < 1 + 1/n$ for every $n\in \mathbb{N}$. The secret to guarantee the Lipschitz constant for the ...

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Why can the first term of the Taylor series expansion of $\cos(\theta)$ be written in the way below?
0 votes

As @lulu said in his comment, the linear approximation or a first order (order $n=1$) Taylor approximation of a function $f$ around $x_0$ is $$f(x)\approx f(x_0)+(x-x_0)f'(x_0),$$ since the Taylor ...

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If a function $f(x)$ is Riemann integrable on $[a,b]$ and $|f(x)| \le M$ for $x \in [a, b]$, show $|\int_a^b f(x)dx| \le M(b - a).$
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Hint: If you think about it, this comes down to prove that $$\left|\int_a^b f(x) \mathrm{d}x\right|\leq \int_a^b |f(x)|\mathrm{d}x.$$ To help you, let us prove that $|f|:x\mapsto |f(x)|$ is ...

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Prove that $f$ is Riemann integrable on $[0,1]$, and find $\int_0^1f(x)dx$
0 votes

It does not matter if it is decreasing. Given the set $A=\{1/2,...,1/n\}$, define your partition $P=\{x_0,x_1...,x_{2n+2}\}$ such that for every $k\in\{1,...,n\}$, $$\frac{1}{k}-\frac{\varepsilon}{2n}\...

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Minimum value of a continuous function, only defined for positive x
0 votes

For the function $f$, you are right. For $g$, the result comes from this theorem: Let $f:X\rightarrow \mathbb{R}$ be a continuous function. If $X$ is compact, then $f(X)$ is also compact. ...

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How is tan φ = (dy/dθ) / (dx/dθ)?
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You can easily see that the tangent of a angle $\phi$ of a parametrized curve $(y(t),x(t))$, at $t=\theta$, with respect to the $x$-axis is \begin{equation}\tan{\phi} = \lim_{t\rightarrow \theta} \...

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