Utilizador:Dante's Imp

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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.

Latex Lab

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

Peirce Notation

${}_{0}^{1}{\overset {11}{\underset {00}{\mathrm {X} }}}{}_{1}^{0}$

Conjunto das partes

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

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

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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• • Para $n=1\,\!$

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

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

$1=1\,\!$

• • Hipótese:

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

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

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

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

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

Teoria das Magnitudes

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

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

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

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

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

---

$B\to A$

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

$B\to \neg A$

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

$\neg B\to A$

$A\lor B$

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

$\neg B\to \neg A$

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

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

$\neg A\land B$

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

$A\land B$

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

$\neg A\land \neg B$

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

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

$A\land \neg B$

1. $\forall x\;\forall y\;\left(x+sy=s\left(x+y\right)\right)$
2. $\forall x\;\left(x+0=x\right)$
3. $\forall y\;\left(s0+sy=s\left(s0+y\right)\right)$
4. $s0+s0=s\left(s0+0\right)$
5. $s0+0=s0\,\!$
6. $s0+s0=ss0\,\!$

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

$G\emptyset D$

$G\varnothing D$

Fórmulas para os tablôs

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

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

$\mathbf {F} \quad B$

$\mathbf {F} \quad A$

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

$\mathbf {F} \quad A$

$\mathbf {F} \quad B$

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

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

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

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

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

Mr. Owl w\ Open Wings