Usuário:Dante's Imp

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Este usuário é um sock puppet
do Wikilivros

Esta conta é um Sock Puppet do usuário Dante Cardoso Pinto de Almeida, e todas suas ações são responsabilidade dele. Como um Sock Puppet, essa conta não pode participar de votações e não pode ser eleita a nenhum cargo.

Currículo Lattes de Dante Cardoso Pinto de Almeida

Filomatia - Página sobre assuntos diversos como Filosofia, Matemática, Lógica, Programação etc.

[editar] Latex Lab

$f(x) = (x^2=1) = \begin{cases} verdade, & \mbox{se }x=1 \,\,\mbox{ou}\,\, x=-1\\ falso, & \mbox{se }x \neq 1\,\,\mbox{e}\,\, x \neq -1 \end{cases}$

[editar] Conjunto das partes

$A=\left\{\heartsuit ,\ \clubsuit ,\ \spadesuit \right\}$

$\mathcal{P}\left(A\right)=\left\{\left\{\heartsuit ,\ \clubsuit ,\ \spadesuit \right\} , \left\{\heartsuit ,\ \clubsuit \right\} , \left\{\heartsuit ,\ \spadesuit \right\} , \left\{\clubsuit ,\ \spadesuit \right\} , \left\{\heartsuit \right\}, \left\{\clubsuit \right\} , \left\{\spadesuit \right\}, \left\{\right\} \right\}$

[editar] Provas por indução

$1^2+2^2+3^2+...+n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$

- Para $n=1\,\!$

$1^2=\frac{1\times\left(1+1\right)\left(2\times 1+1\right)}{6}$

$1=\frac{2\times\left(3\right)}{6}$

$1=\frac{6}{6}$

$1=1\,\!$

- Hipótese:

$1^2+2^2+3^2+...+k^2=\frac{k\left(k+1\right)\left(2k+1\right)}{6}$

$1^2+2^2+3^2+...+k^2+\left(k+1\right)^2=\left(k+1\right)^2+\frac{k\left(k+1\right)\left(2k+1\right)}{6}$

$=k^2+2k+1+\frac{\left(k^2+k\right)\left(2k+1\right)}{6}$

$=\frac{6k^2+12k+6+\left(k^2+k\right)\left(2k+1\right)}{6}$

$=\frac{6k^2+12k+6+2k^3+k^2+2k^2+k}{6}$

$=\frac{2k^3+9k^2+13k+6}{6}$

$=\frac{2k^2\left(k+1\right)+7k^2+13\left(k+1\right)-7}{6}$

$=\frac{2k^2\left(k+1\right)+13\left(k+1\right)+7\left(k^2-1\right)}{6}$

$=\left(k+1\right)\frac{2k^2+13+7\left(k-1\right)}{6}$

$=\left(k+1\right)\frac{2k^2+7k+6}{6}$

$=\left(k+1\right)\frac{\left(k+2\right)\left(2k+3\right)}{6}$

$1^3+2^3+3^3+...+n^3=\frac{n^2\left(n+1\right)^2}{4}$

- Para $n=1\,\!$

$1^3=\frac{1^2\left(1+1\right)^2}{4}$

$1=\frac{4}{4}$

$1=1\,\!$

- Hipótese:

$1^3+2^3+3^3+...+k^3=\frac{k^2\left(k+1\right)^2}{4}$

$1^3+2^3+3^3+...+k^3+\left(k+1\right)^3=\left(k+1\right)^3+\frac{k^2\left(k+1\right)^2}{4}$

$=k^3+3k^2+3k+1+\frac{k^2\left(k^2+2k+1\right)}{4}$

$=\frac{4k^3+12k^2+12k+4+k^4+2k^3+k^2}{4}$

$=\frac{k^4+6k^3+13k^2+12k+4}{4}$

[editar] Teoria das Magnitudes

Ax1. $A<B\Rightarrow \neg\left(B<A\right)$

Ax2. $A<B \and B<C \Rightarrow \neg\left(A<C\right)$

Ax3. $A\ne B \Rightarrow A<B \lor B<A$

Def1. $A>B\ \overset{\underset{\mathrm{def}}{}}{=}\ B<A$

Def2. $A=B\ \overset{\underset{\mathrm{def}}{}}{=}\ \neg\left(A<B\right)\and \neg \left(B<A\right)$

$B\to A$

$\neg\left(\neg A\land B\right)$

$B\to \neg A$

$\neg\left(A\land B\right)$

$\neg B \to A$

$A\lor B$

$\neg \left(\neg A \land \neg B\right)$

$\neg B\to \neg A$

$\neg \left(A\land \neg B\right)$

$\neg \left(B\to A\right)$

$\neg A\land B$

$\neg \left(B\to \neg A\right)$

$A \land B$

$\neg \left(\neg B\to A\right)$

$\neg A \land \neg B$

$\neg \left(A\lor B\right)$

$\neg \left(\neg B\to \neg A\right)$

$A \land \neg B$

$\forall x\;\forall y\; \left (x+sy=s\left (x+y\right )\right )$
$\forall x\; \left (x+0=x\right )$
$\forall y\; \left (s0+sy=s\left (s0+y\right )\right )$
$s0+s0=s\left (s0+0\right )$
$s0+0=s0\,\!$
$s0+s0=ss0 \,\!$

$1=1\,\!$
$\exists x\left(x=1\right)$

$G\empty D$

$G\varnothing D$

[editar] Fórmulas para os tablôs

$\mathbf{V}\quad \neg\left(A\land B\right)$

$\mathbf{V}\quad \neg A$

$\mathbf{F}\quad B$

$\mathbf{F}\quad A$

$\mathbf{F}\quad A\land B$

$\mathbf{F}\quad A$

$\mathbf{F}\quad B$

[editar] Regras derivadas

Dupla Negação (DN)

$\frac{\neg \neg \alpha\,}{\overline{\alpha\quad\;\,}}$

Silogismo Hipotético (SH)

$\alpha \to \beta\,\!$

$\underline{\beta \to \gamma}\,\!$

$\alpha \to \gamma\,\!$

Repetição (R)

$\frac{\alpha}{\alpha} \,\!$

Modus Tollens (MT)

$\alpha \to \beta\,\!$

$\underline{\neg \beta \quad\ }\,\!$

$\neg \alpha\,\!$

Prefixação (PRF)

$\frac{\alpha}{\beta \to \alpha}$

Contraposição (CT)

$\frac{\alpha \to \beta}{\overline{\neg \beta\to \neg \alpha}}$

Contradição (CTR)

$\alpha \,\!$

$\underline{\neg \alpha \ }\,\!$

$\beta \,\!$

Lei de Duns Scot

$\frac{\neg \alpha}{\alpha \to \beta}$

$\frac{\alpha}{\neg \alpha \to \beta}$

Leis DeMorgan

$\frac{\neg \left( \alpha \lor \beta \right)}{\overline{\,\neg \alpha \land \neg \beta}}$

$\frac{\neg \left(\alpha\land \beta\right)}{\overline{\neg \alpha \lor \neg \beta\ }}$

[editar] Latex Art Lab

(\__/)
(O.o)
(> <)

$\left( \setminus\underline{\quad} /\right)$
$\left( O\cdot o\right)$
$\left( >\, <\right)$

Mr. Bunny Interested

${\color{White}\cdot}\!\!\land\!\underline{ \ }\land$
$\left( 0_\lor 0\right)$
${\color{White}\cdot}\!\!\!\left( >\, <\right)$

$Oh, Rly?\,\!$

Mr. Owl

${\color{White}\cdot}\,\cdot\!\!\bigcap\!\!\cdot$
${\color{White}\cdot}\!\!^\subset_\subset\!\!\!\left(\overset{\ \,}{\underset{\ }{\ }}\ \right)\!\!\! ^\supset_\supset$
${\color{White}\cdot\cdot}\ \ \!\lor$

Mr. Tortoise

${\color{White}\cdot}\!\!\vartriangle\vartriangle$
$\left(\bullet\bullet\right)$
${\color{White}\cdot}\!\!\left( \ \, _\pitchfork\right)\!\!_\rightsquigarrow$

Mr. Imp

${\color{White}\cdot}\!\!\!\left( \setminus\underline{\quad} /\right)$
$\left( \neg_{\blacktriangle} \neg\right)$
${\color{White}\cdot}\left(\lor\, \lor\right)$

Mr. Bunny Suspicious

${\color{White}\cdot}\!\!\!\left( \setminus\underline{\quad} /\right)$
$\left( ^0 \blacktriangle ^0\right)$
${\color{White}\cdot}\left(\lor\, \lor\right)$

Mr. Bunny Surprised

${\color{White}\cdot}\!\!\!\left( \setminus\underline{\quad} /\right)$
${\color{White}\cdot}\!\!\left( \circledast _\blacktriangle \circledast\right)$
${\color{White}\cdot}\left(\lor\, \lor\right)$

Mr. Bunny Addicted

${\color{White}\cdot}\!\!\!\circ\quad \!\circ$
$\left( ^\bullet \blacktriangle ^\bullet\right)$
${\color{White}\cdot}\!\!\left(>\, <\right)$

Mr. Coala

${\color{White}\cdot}{\color{White}\cdot}\!\land\!\underline{ \ }\land$
${\color{White}\cdot}\left( 0_\lor 0\right)$
${\color{White}\cdot}\!\!\!<\!\!\!\left( \quad \ \right)\!\!\!>$