Análisis Funcional y Lógica Algebraica y Lógica Dualógica

( A[z] = x & B[z] = y ) <==> [Et][ A[t] [o] B[t] = 0 ]

A[z] = z+ia & B[z] = z+(-i)·a

( t = (-i)·a || t = ia )

A[(-i)·a] = 0 & B[ia] = 0

A[z] = z^{n}+i & B[z] = z^{n}+(-i)

( t = e^{(-1)·(1/2)·(1/n)·pi·i} || t = e^{(1/2)·(1/n)·pi·i} )

A[e^{(-1)·(1/2)·(1/n)·pi·i}] = 0 & B[e^{(1/2)·(1/n)·pi·i}] = 0

A[z] = a+iz & B[z] = a+(-i)·z

( t = ia || t = (-i)·a )

A[ia] = 0 & B[(-i)·a] = 0

A[z] = int[z+i]d[z] & B[z] = int[z+(-i)]d[z]

( t = 2·(-i) || t = 2i )

A[(-i)] = 0 & B[i] = 0

Hablash lu portuguese-y que yo hablu-puesh?

Hablu-puesh lu portuguese-y que tú hablash.

Voy-puesh a cojer un resfriadu

porque quishase-y no coju-puesh una chaqueta-y,

cuandu estamush en hinviernu.

No voy-puesh a cojer un resfriadu

aunque no coju-puesh una chaqueta-y,

cuandu estamush en veranu.

En recuerdo de mi amigo Hugo matemático portugués:

Definición:

[f(x)] <==> Concreto

]f(x)[ <==> Semejante-Abstracto

Lógica algebraica de suma:

Teorema:

[f(1)]+[g(1)] = ]h(2)[

[f(1)]+]g(n)[ = ]h(n+1)[

Demostración:

1+1 = 2

Si n != 0 ==> n+1 != 1

Teorema:

[f(-1)]+[g(-1)] = ]h(-2)[

[f(-1)]+]g(-n)[ = ]h((-1)·(n+1))[

Demostración:

(-1)+(-1) = (-2)

Si (-n) != (-0) ==> (-1)·(n+1) != (-1)

Teorema

Sea f(x) una función concreta.

Concreto en f(x)

Semejante-Abstracto en n·f(x)

Demostración:

[f(x)]+[f(x)] = ](1+1)·f(x)[ = ]2·f(x)[

[f(x)]+]n·f(x)[ = ](n+1)·f(x)[

Teorema:

Sea f(x) una función concreta.

Concreto en (-1)·f(x)

Semejante-Abstracto en (-n)·f(x)

Demostración:

[(-1)·f(x)]+[(-1)·f(x)] = ]((-1)+(-1))·f(x)[ = ](-2)·f(x)[

[(-1)·f(x)]+](-n)·f(x)[ = ]((-n)+(-1))·f(x)[ = ](-1)·(n+1)·f(x)[

Concreto en circunferencia elíptica:

[sin(x)]+[sin(x)] = ]sin[2](x)[

[sin(x)]+]sin[n](x)[ = ]sin[n+1](x)[

[cos(x)]+[cos(x)] = ]cos[2](x)[

[cos(x)]+]cos[n](x)[ = ]cos[n+1](x)[

Concreto en circunferencia hiperbólica:

[sinh(x)]+[sinh(x)] = ]sinh[2](x)[

[sinh(x)]+]sinh[n](x)[ = ]sinh[n+1](x)[

[cosh(x)]+[cosh(x)] = ]cosh[2](x)[

[cosh(x)]+]cosh[n](x)[ = ]cosh[n+1](x)[

Concreto en exponente 1:

[x]+[x] = ]x^{2}[

[x]+]x^{n}[ = ]x^{n+1}[

Concreto en exponente (-1):

[(1/x)]+[(1/x)] = ](1/x^{2})[

[(1/x)]+](1/x^{n})[ = ](1/x^{n+1})[

Concreto en d_{x}[e^{x}] = e^{x}

[e^{x}]+[e^{x}] = ]e^{2x}[

[e^{x}]+]e^{nx}[ = ]e^{(n+1)·x}[

Concreto en d_{x}[e^{(-x)}] = (-1)·e^{(-x)}

[e^{(-x)}]+[e^{(-x)}] = ]e^{(-2)·x}[

[e^{(-x)}]+]e^{(-n)·x}[ = ]e^{(-1)·(n+1)·x}[

Concreto en 1:

[< 1,...(k)...,1 >]+[< 1,...(k)...,1 >] = ]< 2,...(k)...,2 >[

[< 1,...(k)...,1 >]+]< n,...(k)...,n >[ = ]< (n+1),...(k)...,(n+1) >[

Concreto en (-1):

[< (-1),...(k)...,(-1) >]+[< (-1),...(k)...,(-1) >] = ]< (-2),...(k)...,(-2) >[

[< (-1),...(k)...,(-1) >]+]< (-n),...(k)...,(-n) >[ = ]< (-1)·(n+1),...(k)...,(-1)·(n+1) >[

Concreto en [Ai][ 1_{i...(k)...i} = 1 ]

Semejante-Abstracto en [Ai][ n_{i...(k)...i} = n ]

[1_{i...(k)...i}]+[1_{i...(k)...i}] = ](1+1)_{i...(k)...i}[ = ]2_{i...(k)...i}[

[1_{i...(k)...i}]+]n_{i...(k)...i}[ = ](n+1)_{i...(k)...i}[

Concreto en [Ai][ (-1)_{i...(k)...i} = (-1) ]

Semejante-Abstracto en [Ai][ (-n)_{i...(k)...i} = (-n) ]

[(-1)_{i...(k)...i}]+[(-1)_{i...(k)...i}] = ]((-1)+(-1))_{i...(k)...i}[ = ](-2)_{i...(k)...i}[

[(-1)_{i...(k)...i}]+](-n)_{i...(k)...i}[ = ]((-n)+(-1))_{i...(k)...i}[ = ]((-1)·(n+1))_{i...(k)...i}[

Concreto en:

( < 1,0,..(k)..,0,1 >,< 0,0,..(k)..,0,0 >,..(k)..,< 0,0,..(k)..,0,0 >,< 1,0,..(k)..,0,1 > )

Semejante-Abstracto en:

( < n,0,..(k)..,0,n >,< 0,0,..(k)..,0,0 >,..(k)..,< 0,0,..(k)..,0,0 >,< n,0,..(k)..,0,n > )

Concreto en:

( < (-1),0,..(k)..,0,(-1) >,< 0,0,..(k)..,0,0 >,..(k)..,< 0,0,..(k)..,0,0 >,< (-1),0,..(k)..,0,(-1) > )

Semejante-Abstracto en:

( < (-n),0,..(k)..,0,(-n) >,< 0,0,..(k)..,0,0 >,..(k)..,< 0,0,..(k)..,0,0 >,< (-n),0,..(k)..,0,(-n) > )

Concreto en [Ej_{0}][Ai][ 1_{i}^{j_{0}} = 1 ] 

Semejante-Abstracto en [Ej_{0}][Ai][ n_{i}^{j_{0}} = n ]

Concreto en [Ej_{0}][Ai][ (-1)_{i}^{j_{0}} = (-1) ] 

Semejante-Abstracto en [Ej_{0}][Ai][ (-n)_{i}^{j_{0}} = (-n) ]

Concreto en [Ei_{0}][Aj][ 1_{i_{0}}^{j} = 1 ] 

Semejante-Abstracto en [Ei_{0}][Aj][ n_{i_{0}}^{j} = n ]

Concreto en [Ei_{0}][Aj][ (-1)_{i_{0}}^{j} = (-1) ] 

Semejante-Abstracto en [Ei_{0}][Aj][ (-n)_{i_{0}}^{j} = (-n) ]

Lógica algebraica de conectivas:

Teorema:

min{[f(x)],]g(x)[} = ]h(x)[ <==> max{[f(x)],]g(x)[} = [h(x)]

max{¬[f(x)],¬]g(x)[} = ]h(x)[ <==> min{¬[f(x)],¬]g(x)[} = [h(x)]

Demostración:

min = ]g(x)[ <==> max = [f(x)]

max = ]¬g(x)[ <==> min = [¬f(x)]

Concreto en binario:

min{1,(2/3)} = (2/3) <==> max{1,(2/3)} = 1

max{0,(1/3)} = (1/3) <==> min{0,(1/3)} = 0

Concreto en binario:

min{1,(3/4)} = (3/4) <==> max{1,(3/4)} = 1

max{0,(1/4)} = (1/4) <==> min{0,(1/4)} = 0

Concreto en exponente 1 y en exponente (-1):

max{[x],]x^{n}[} = ]x^{n}[ <==> min{[x],]x^{n}[} = [x]

min{[(1/x)],](1/x^{n})[} = ](1/x^{n})[ <==> max{[(1/x)],](1/x^{n})[} = [(1/x)]

Teorema:

max{¬[f(x)],]g(x)[} = ]h(x)[ <==> min{¬[f(x)],]g(x)[} = [h(x)]

min{[f(x)],¬]g(x)[} = ]h(x)[ <==> max{[f(x)],¬]g(x)[} = [h(x)]

Teorema:

max{min{[f(x)],¬]g(x)[},min{¬[f(x)],]g(x)[}} = ]h(x)[ <==> ...

... min{min{[f(x)],¬]g(x)[},min{¬[f(x)],]g(x)[}} = [h(x)]

min{max{[f(x)],¬]g(x)[},max{¬[f(x)],]g(x)[}} = ]h(x)[ <==> ...

... max{max{[f(x)],¬]g(x)[},max{¬[f(x)],]g(x)[}} = [h(x)]

Lógica algebraica dualógica:

Teorema:

[f(1)]+[g(0)] = [h(1)]

]f((n+(-1))/n)[+]g(1/n)[ = [h(1)]

Demostración:

1+0 = 1

((n+(-1))/n)+(1/n) = (n/n) = 1

Teorema:

[f(-1)]+[g(-0)] = [h(-1)]

]f((-1)·((n+(-1))/n))[+]g((-1)·(1/n))[ = [h(-1)]

Demostración:

(-1)+(-0) = (-1)

(-1)·((n+(-1))/n)+(-1)·(1/n) = (-1)·(n/n) = (-1)

Teorema:

Sea f(z) una función concreta.

[x]+[y] = [f(z)] <==> ( x = f(z) & y = 0 )

]x[+]y[ = [f(z)] <==> ( x = f(z)+(-1)·(f(z)/n) & y = (f(z)/n) )

Teorema:

Sea f(z) una función concreta.

[x]+[y] = [(-1)·f(z)] <==> ( x = (-1)·f(z) & y = (-0) )

]x[+]y[ = [(-1)·f(z)] <==> ( x = (-1)·f(z)+(f(z)/n) & y = (-1)·(f(z)/n) )

[x]+[y] = [1] <==> ( x = 1 & y = 0 )

]x[+]y[ = [1] <==> ( x = 1+(-1)·(1/n) & y = (1/n) )

[x]+[y] = [(-1)] <==> ( x = (-1) & y = (-0) )

]x[+]y[ = [(-1)] <==> ( x = (-1)+(1/n) & y = (-1)·(1/n) )

[x]+[y] = [z] <==> ( x = z & y = 0 )

]x[+]y[ = [z] <==> ( x = z+(-1)·(z/n) & y = (z/n) )

[x]+[y] = [(1/z)] <==> ( x = (1/z) & y = 0 )

]x[+]y[ = [(1/z)] <==> ( x = (1/z)+(-1)·(1/(nz)) & y = (1/(nz)) )

[x]+[y] = [e^{z}] <==> ( x = e^{z} & y = 0 )

]x[+]y[ = [e^{z}] <==> ( x = e^{z}+(-1)·(e^{z}/n) & y = (e^{z}/n) )

[x]+[y] = [e^{(-z)}] <==> ( x = e^{(-z)} & y = 0 )

]x[+]y[ = [e^{(-z)}] <==> ( x = e^{(-z)}+(-1)·(e^{(-z)}/n) & y = (e^{(-z)}/n) )

Lógica Dualógica:

Definición:

Dual-Conectivo = { < x,y > : v( G(x,y) <==> F(x,y) ) = 1 }

Anti-Dual-Conectivo = { < x,y > : v( G(x,y) <==> F(x,y) ) = 0 }

Teorema:

( G(x,y) <==> F(x,y) ) es dualogía.

Demostración:

g(x,y) = f(x)

G(x,y) <==> F(x,y)

g(a,0) = f(a)

G(a,0) <==> F(a,0)

g(b,1) = f(b)

G(b,1) <==> F(b,1)

( f(x) = x+(-1) || f(x) = x || f(x) = x+1 )

( F(x,y) <==> x+(-y) || F(x,y) <==> x+(-y)+1 || F(x,y) <==> x+y || F(x,y) <==> x+y+(-1) )

( g(x,y) = x+(-y) || g(x,y) = x+(-y)+1 || g(x,y) = x+y || g(x,y) = x+y+(-1) )

( G(x,y) <==> x+(-y) || G(x,y) <==> x+(-y)+1 || G(x,y) <==> x+y || G(x,y) <==> x+y+(-1) )

f(x) es biyectiva

F(x,0) es biyectiva.

F(x,1) es biyectiva

g(x,0) es biyectiva.

g(x,1) es biyectiva

G(x,0) es biyectiva.

G(x,1) es biyectiva

< h: A ---> B & a ---> h(F(a,0)) = f(a) >

< H: A ---> B & a ---> H(f(a)) <==> F(a,1) >

h(F(a,0)) = h(F(c,0))

f(a) = f(c)

a = c

< a,0 > = < c,0 >

F(a,0) <==> F(c,0)

H(f(a)) <==> H(f(c))

< h: A ---> B & b ---> h(F(b,1)) = f(b) >

< H: A ---> B & a ---> H(f(b)) <==> F(b,1) >

h(F(b,1)) = h(F(d,1))

f(b) = f(d)

b = d

< b,1 > = < d,1 >

F(b,1) <==> F(d,1)

H(f(b)) <==> H(f(d))

< h: A ---> B & a ---> h(G(a,0)) = g(a,0) >

< H: A ---> B & a ---> H(g(a,0)) <==> G(a,1) >

h(G(a,0)) = h(G(c,0))

g(a,0) = g(c,0)

< a,0 > = < c,0 >

G(a,0) <==> G(c,0)

H(g(a,0)) <==> H(g(c,0))

< h: A ---> B & b ---> h(F(b,1)) = f(b) >

< H: A ---> B & a ---> H(f(b)) <==> F(b,1) >

h(G(b,1)) = h(G(d,1))

g(b,1) = g(d,1)

< b,1 > = < d,1 >

G(b,1) <==> G(d,1)

H(g(b,1)) <==> H(g(d,1))

( x(t) & y(t) ) <==> 1 ...

... dual[&] = { < 1,1 > }

( x(t) || y(t) ) <==> 1 ...

... dual[ || ] = { < 1,1 >,< 1,0 >,< 0,1 > }

( x(t) || y(t) ) <==> 0 ...

... dual[ || ] = { < 0,0 > }

( x(t) & y(t) ) <==> 0 ...

... dual[&] = { < 0,0 >,< 0,1 >,< 1,0 > }

( x(t) & y(t) ) <==> ( x(t) <==> y(t) ) ...

... dual[&] = { < 1,1 > }

( x(t) || y(t) ) <==> ( x(t) <==> y(t) ) ...

... dual[ || ] = { < 1,1 > }

¬( x(t) & y(t) ) <==> ( x(t) <==> y(t) ) ...

... dual[&] = { < 0,0 > }

¬( x(t) || y(t) ) <==> ( x(t) <==> y(t) ) ...

... dual[ || ] = { < 0,0 > }

( x(t) || y(t) ) <==> ( x(t) ==> y(t) ) ...

... dual[ || ] = { < 1,1 >,< 0,1 > }

( x(t) || y(t) ) <==> ( x(t) <== y(t) ) ...

... dual[ || ] = { < 1,1 >,< 1,0 > }

¬( x(t) & y(t) ) <==> ( x(t) <== y(t) ) ...

... dual[&] = { < 0,0 >,< 1,0 > }

¬( x(t) & y(t) ) <==> ( x(t) ==> y(t) ) ...

... dual[&] = { < 0,0 >,< 0,1 > }

( x(t) & y(t) ) <==> ( x(t) |o| y(t) ) ...

... dual[&] = { < 0,0 > }

( x(t) || y(t) ) <==> ( x(t) |o| y(t) ) ...

... dual[ || ] = { < 0,0 >,< 1,0 >,< 0,1 > }

¬( x(t) || y(t) ) <==> ( x(t) |o| y(t) ) ...

... dual[ || ] = { < 1,1 > }

¬( x(t) & y(t) ) <==> ( x(t) |o| y(t) ) ...

... dual[&] = { < 1,1 >,< 0,1 >,< 1,0 > }

Laboratorio de Problemas:

( x(t) & y(t) ) <==> ( x(t) ==> y(t) )

( x(t) & y(t) ) <==> ( x(t) <== y(t) )

¬( x(t) || y(t) ) <==> ( x(t) <== y(t) )

¬( x(t) || y(t) ) <==> ( x(t) ==> y(t) )

Máster en Lógica Algebraica y Dualogía:

Lógica Algebraica de Suma.

Lógica Algebraica de Conectiva.

Lógica Algebraica Dualógica.

Lógica Dualógica.

Álgebra Dualógica.

Análisis Dualógico.

Análisis Dualógico:

(-1) [< x+y [< x+(-a)

(-1)+(-a) [< y [< (-a)

dual[x+(-a)] = { < a,(-a)+(-1)·(1/n) > }

1 >] x+y >] x+a

1+a >] y >] a

dual[x+a] = { < (-a),a+(1/n) > }

(-m) [< x+y [< x^{m}+(-a)

(-m)+(-1)·a^{(1/m)}·e^{(2/m)·pi·i} [< y [< (-1)·a^{(1/m)}·e^{(2/m)·pi·i}

dual[x^{m}+(-a)] = ...

... { < a^{(1/m)}·e^{(2/m)·pi·i},(-1)·a^{(1/m)}·e^{(2/m)·pi·i}+(-1)·(m/n) > }

m >] x+y >] x^{m}+a

m+(-1)·a^{(1/m)}·e^{(1/m)·pi·i} >] y >] (-1)·a^{(1/m)}·e^{(1/m)·pi·i}

dual[x^{m}+a] = ...

... { < a^{(1/m)}·e^{(1/m)·pi·i},(-1)·a^{(1/m)}·e^{(1/m)·pi·i}+(m/n) > }

(-m) [< x+y [< (x+(-a))^{m}

(-m)+(-a) [< y [< (-a)

dual[(x+(-a))^{m}] = { < a,(-a)+(-a)·(m/n) > }

m >] x+y >] (x+a)^{m}

m+a >] y >] a

dual[(x+a)^{m}] = { < (-a),a+(m/n) > }