f
(
x
)
=
(
x
2
=
1
)
=
{
v
e
r
d
a
d
e
,
se
x
=
1
ou
x
=
−
1
f
a
l
s
o
,
se
x
≠
1
e
x
≠
−
1
{\displaystyle 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
editar
0
1
X
00
11
1
0
{\displaystyle {}_{0}^{1}{\overset {11}{\underset {00}{\mathrm {X} }}}{}_{1}^{0}}
Conjunto das partes
editar
A
=
{
♡
,
♣
,
♠
}
{\displaystyle A=\left\{\heartsuit ,\ \clubsuit ,\ \spadesuit \right\}}
P
(
A
)
=
{
{
♡
,
♣
,
♠
}
,
{
♡
,
♣
}
,
{
♡
,
♠
}
,
{
♣
,
♠
}
,
{
♡
}
,
{
♣
}
,
{
♠
}
,
{
}
}
{\displaystyle {\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
editar
1
2
+
2
2
+
3
2
+
.
.
.
+
n
2
=
n
(
n
+
1
)
(
2
n
+
1
)
6
{\displaystyle 1^{2}+2^{2}+3^{2}+...+n^{2}={\frac {n\left(n+1\right)\left(2n+1\right)}{6}}}
Para
n
=
1
{\displaystyle n=1\,\!}
1
2
=
1
×
(
1
+
1
)
(
2
×
1
+
1
)
6
{\displaystyle 1^{2}={\frac {1\times \left(1+1\right)\left(2\times 1+1\right)}{6}}}
1
=
2
×
(
3
)
6
{\displaystyle 1={\frac {2\times \left(3\right)}{6}}}
1
=
6
6
{\displaystyle 1={\frac {6}{6}}}
1
=
1
{\displaystyle 1=1\,\!}
1
2
+
2
2
+
3
2
+
.
.
.
+
k
2
=
k
(
k
+
1
)
(
2
k
+
1
)
6
{\displaystyle 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
+
(
k
+
1
)
2
=
(
k
+
1
)
2
+
k
(
k
+
1
)
(
2
k
+
1
)
6
{\displaystyle 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
+
2
k
+
1
+
(
k
2
+
k
)
(
2
k
+
1
)
6
{\displaystyle =k^{2}+2k+1+{\frac {\left(k^{2}+k\right)\left(2k+1\right)}{6}}}
=
6
k
2
+
12
k
+
6
+
(
k
2
+
k
)
(
2
k
+
1
)
6
{\displaystyle ={\frac {6k^{2}+12k+6+\left(k^{2}+k\right)\left(2k+1\right)}{6}}}
=
6
k
2
+
12
k
+
6
+
2
k
3
+
k
2
+
2
k
2
+
k
6
{\displaystyle ={\frac {6k^{2}+12k+6+2k^{3}+k^{2}+2k^{2}+k}{6}}}
=
2
k
3
+
9
k
2
+
13
k
+
6
6
{\displaystyle ={\frac {2k^{3}+9k^{2}+13k+6}{6}}}
=
2
k
3
+
9
k
2
+
13
k
+
6
6
{\displaystyle ={\frac {2k^{3}+9k^{2}+13k+6}{6}}}
=
2
k
2
(
k
+
1
)
+
7
k
2
+
13
(
k
+
1
)
−
7
6
{\displaystyle ={\frac {2k^{2}\left(k+1\right)+7k^{2}+13\left(k+1\right)-7}{6}}}
=
2
k
2
(
k
+
1
)
+
13
(
k
+
1
)
+
7
(
k
2
−
1
)
6
{\displaystyle ={\frac {2k^{2}\left(k+1\right)+13\left(k+1\right)+7\left(k^{2}-1\right)}{6}}}
=
(
k
+
1
)
2
k
2
+
13
+
7
(
k
−
1
)
6
{\displaystyle =\left(k+1\right){\frac {2k^{2}+13+7\left(k-1\right)}{6}}}
=
(
k
+
1
)
2
k
2
+
7
k
+
6
6
{\displaystyle =\left(k+1\right){\frac {2k^{2}+7k+6}{6}}}
=
(
k
+
1
)
(
k
+
2
)
(
2
k
+
3
)
6
{\displaystyle =\left(k+1\right){\frac {\left(k+2\right)\left(2k+3\right)}{6}}}
=
(
k
+
1
)
(
k
+
1
+
1
)
(
2
(
k
+
1
)
+
1
)
6
{\displaystyle ={\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
=
n
2
(
n
+
1
)
2
4
{\displaystyle 1^{3}+2^{3}+3^{3}+...+n^{3}={\frac {n^{2}\left(n+1\right)^{2}}{4}}}
Para
n
=
1
{\displaystyle n=1\,\!}
1
3
=
1
2
(
1
+
1
)
2
4
{\displaystyle 1^{3}={\frac {1^{2}\left(1+1\right)^{2}}{4}}}
1
=
4
4
{\displaystyle 1={\frac {4}{4}}}
1
=
1
{\displaystyle 1=1\,\!}
1
3
+
2
3
+
3
3
+
.
.
.
+
k
3
=
k
2
(
k
+
1
)
2
4
{\displaystyle 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
+
(
k
+
1
)
3
=
(
k
+
1
)
3
+
k
2
(
k
+
1
)
2
4
{\displaystyle 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
+
3
k
2
+
3
k
+
1
+
k
2
(
k
2
+
2
k
+
1
)
4
{\displaystyle =k^{3}+3k^{2}+3k+1+{\frac {k^{2}\left(k^{2}+2k+1\right)}{4}}}
=
4
k
3
+
12
k
2
+
12
k
+
4
+
k
4
+
2
k
3
+
k
2
4
{\displaystyle ={\frac {4k^{3}+12k^{2}+12k+4+k^{4}+2k^{3}+k^{2}}{4}}}
=
k
4
+
6
k
3
+
13
k
2
+
12
k
+
4
4
{\displaystyle ={\frac {k^{4}+6k^{3}+13k^{2}+12k+4}{4}}}
Teoria das Magnitudes
editar
Ax1.
A
<
B
⇒
¬
(
B
<
A
)
{\displaystyle A<B\Rightarrow \neg \left(B<A\right)}
Ax2.
A
<
B
∧
B
<
C
⇒
¬
(
A
<
C
)
{\displaystyle A<B\land B<C\Rightarrow \neg \left(A<C\right)}
Ax3.
A
≠
B
⇒
A
<
B
∨
B
<
A
{\displaystyle A\neq B\Rightarrow A<B\lor B<A}
Def1.
A
>
B
=
d
e
f
B
<
A
{\displaystyle A>B\ {\overset {\underset {\mathrm {def} }{}}{=}}\ B<A}
Def2.
A
=
B
=
d
e
f
¬
(
A
<
B
)
∧
¬
(
B
<
A
)
{\displaystyle A=B\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \neg \left(A<B\right)\land \neg \left(B<A\right)}
B
→
A
{\displaystyle B\to A}
¬
(
¬
A
∧
B
)
{\displaystyle \neg \left(\neg A\land B\right)}
B
→
¬
A
{\displaystyle B\to \neg A}
¬
(
A
∧
B
)
{\displaystyle \neg \left(A\land B\right)}
¬
B
→
A
{\displaystyle \neg B\to A}
A
∨
B
{\displaystyle A\lor B}
¬
(
¬
A
∧
¬
B
)
{\displaystyle \neg \left(\neg A\land \neg B\right)}
¬
B
→
¬
A
{\displaystyle \neg B\to \neg A}
¬
(
A
∧
¬
B
)
{\displaystyle \neg \left(A\land \neg B\right)}
¬
(
B
→
A
)
{\displaystyle \neg \left(B\to A\right)}
¬
A
∧
B
{\displaystyle \neg A\land B}
¬
(
B
→
¬
A
)
{\displaystyle \neg \left(B\to \neg A\right)}
A
∧
B
{\displaystyle A\land B}
¬
(
¬
B
→
A
)
{\displaystyle \neg \left(\neg B\to A\right)}
¬
A
∧
¬
B
{\displaystyle \neg A\land \neg B}
¬
(
A
∨
B
)
{\displaystyle \neg \left(A\lor B\right)}
¬
(
¬
B
→
¬
A
)
{\displaystyle \neg \left(\neg B\to \neg A\right)}
A
∧
¬
B
{\displaystyle A\land \neg B}
∀
x
∀
y
(
x
+
s
y
=
s
(
x
+
y
)
)
{\displaystyle \forall x\;\forall y\;\left(x+sy=s\left(x+y\right)\right)}
∀
x
(
x
+
0
=
x
)
{\displaystyle \forall x\;\left(x+0=x\right)}
∀
y
(
s
0
+
s
y
=
s
(
s
0
+
y
)
)
{\displaystyle \forall y\;\left(s0+sy=s\left(s0+y\right)\right)}
s
0
+
s
0
=
s
(
s
0
+
0
)
{\displaystyle s0+s0=s\left(s0+0\right)}
s
0
+
0
=
s
0
{\displaystyle s0+0=s0\,\!}
s
0
+
s
0
=
s
s
0
{\displaystyle s0+s0=ss0\,\!}
1
=
1
{\displaystyle 1=1\,\!}
∃
x
(
x
=
1
)
{\displaystyle \exists x\left(x=1\right)}
G
∅
D
{\displaystyle G\emptyset D}
G
∅
D
{\displaystyle G\varnothing D}
Fórmulas para os tablôs
editar
Regras derivadas
editar
¬
¬
α
α
¯
{\displaystyle {\frac {\neg \neg \alpha \,}{\overline {\alpha \quad \;\,}}}}
Silogismo Hipotético (SH)
α
→
β
{\displaystyle \alpha \to \beta \,\!}
β
→
γ
_
{\displaystyle {\underline {\beta \to \gamma }}\,\!}
α
→
γ
{\displaystyle \alpha \to \gamma \,\!}
α
α
{\displaystyle {\frac {\alpha }{\alpha }}\,\!}
α
→
β
{\displaystyle \alpha \to \beta \,\!}
¬
β
_
{\displaystyle {\underline {\neg \beta \quad \ }}\,\!}
¬
α
{\displaystyle \neg \alpha \,\!}
α
β
→
α
{\displaystyle {\frac {\alpha }{\beta \to \alpha }}}
α
→
β
¬
β
→
¬
α
¯
{\displaystyle {\frac {\alpha \to \beta }{\overline {\neg \beta \to \neg \alpha }}}}
α
{\displaystyle \alpha \,\!}
¬
α
_
{\displaystyle {\underline {\neg \alpha \ }}\,\!}
β
{\displaystyle \beta \,\!}
¬
α
α
→
β
{\displaystyle {\frac {\neg \alpha }{\alpha \to \beta }}}
α
¬
α
→
β
{\displaystyle {\frac {\alpha }{\neg \alpha \to \beta }}}
¬
(
α
∨
β
)
¬
α
∧
¬
β
¯
{\displaystyle {\frac {\neg \left(\alpha \lor \beta \right)}{\overline {\,\neg \alpha \land \neg \beta }}}}
¬
(
α
∧
β
)
¬
α
∨
¬
β
¯
{\displaystyle {\frac {\neg \left(\alpha \land \beta \right)}{\overline {\neg \alpha \lor \neg \beta \ }}}}
(\__/) (O.o) (> <)
(
∖
_
/
)
{\displaystyle \left(\setminus {\underline {\quad }}/\right)}
(
O
⋅
o
)
{\displaystyle \left(O\cdot o\right)}
(
>
<
)
{\displaystyle \left(>\,<\right)}
Mr. Bunny Interested
⋅
∧
_
∧
{\displaystyle {\color {White}\cdot }\!\!\land \!{\underline {\ }}\land }
(
0
∨
0
)
{\displaystyle \left(0_{\lor }0\right)}
⋅
(
>
<
)
{\displaystyle {\color {White}\cdot }\!\!\!\left(>\,<\right)}
O
h
,
R
l
y
?
{\displaystyle Oh,Rly?\,\!}
Mr. Owl
⋅
⋅
⋂
⋅
{\displaystyle {\color {White}\cdot }\,\cdot \!\!\bigcap \!\!\cdot }
⋅
⊂
⊂
(
)
⊃
⊃
{\displaystyle {\color {White}\cdot }\!\!_{\subset }^{\subset }\!\!\!\left({\overset {\ \,}{\underset {\ }{\ }}}\ \right)\!\!\!_{\supset }^{\supset }}
⋅
⋅
∨
{\displaystyle {\color {White}\cdot \cdot }\ \ \!\lor }
Mr. Tortoise
⋅
△△
{\displaystyle {\color {White}\cdot }\!\!\vartriangle \vartriangle }
(
∙
∙
)
{\displaystyle \left(\bullet \bullet \right)}
⋅
(
⋔
)
⇝
{\displaystyle {\color {White}\cdot }\!\!\left(\ \,_{\pitchfork }\right)\!\!_{\rightsquigarrow }}
Mr. Imp
⋅
(
∖
_
/
)
{\displaystyle {\color {White}\cdot }\!\!\!\left(\setminus {\underline {\quad }}/\right)}
(
¬
▴
¬
)
{\displaystyle \left(\neg _{\blacktriangle }\neg \right)}
⋅
(
∨
∨
)
{\displaystyle {\color {White}\cdot }\left(\lor \,\lor \right)}
Mr. Bunny Suspicious
⋅
(
∖
_
/
)
{\displaystyle {\color {White}\cdot }\!\!\!\left(\setminus {\underline {\quad }}/\right)}
(
0
▴
0
)
{\displaystyle \left(^{0}\blacktriangle ^{0}\right)}
⋅
(
∨
∨
)
{\displaystyle {\color {White}\cdot }\left(\lor \,\lor \right)}
Mr. Bunny Surprised
⋅
(
∖
_
/
)
{\displaystyle {\color {White}\cdot }\!\!\!\left(\setminus {\underline {\quad }}/\right)}
⋅
(
⊛
▴
⊛
)
{\displaystyle {\color {White}\cdot }\!\!\left(\circledast _{\blacktriangle }\circledast \right)}
⋅
(
∨
∨
)
{\displaystyle {\color {White}\cdot }\left(\lor \,\lor \right)}
Mr. Bunny Addicted
⋅
∘
∘
{\displaystyle {\color {White}\cdot }\!\!\!\circ \quad \!\circ }
(
∙
▴
∙
)
{\displaystyle \left(^{\bullet }\blacktriangle ^{\bullet }\right)}
⋅
(
>
<
)
{\displaystyle {\color {White}\cdot }\!\!\left(>\,<\right)}
Mr. Coala
⋅
⋅
∧
_
∧
{\displaystyle {\color {White}\cdot }{\color {White}\cdot }\!\land \!{\underline {\ }}\land }
⋅
(
0
∨
0
)
{\displaystyle {\color {White}\cdot }\left(0_{\lor }0\right)}
⋅
<
(
)
>
{\displaystyle {\color {White}\cdot }\!\!\!<\!\!\!\left(\quad \ \right)\!\!\!>}
Mr. Owl w\ Open Wings 
