maxim comú divisor álgebra

mcd{x,y} = x [M] y (intersecció en divisors)
mcm{x,y} = x [W] y (reunió en divisors)


mcd{(1/x),(1/y)} = (1/x) [M] (1/y)
mcm{(1/x),(1/y)} = (1/x) [W] (1/y)


mcd{2,6} = 2 [M] 6 = 2
mcm{2,6} = 2 [W] 6 = 6


mcd{(1/2),(1/6)} = (1/2) [M] (1/6) = (1/6)
mcm{(1/2),(1/6)} = (1/2) [W] (1/6) = (1/2)


2·3=6
(1/6)·3=(1/2)


mcd{mcd{x,y},z} =  mcd{x,mcd{y,z}}
mcm{mcm{x,y},z} =  mcm{x,mcm{y,z}}


mcd{mcd{x,y},z} =  mcd{x,mcd{y,z}}
mcd{mcd{x,y},z} = mcd{x,y} [M] z
mcd{mcd{x,y},z} = (x [M] y) [M] z
mcd{mcd{x,y},z} = x [M] (y [M] z)
mcd{mcd{x,y},z} = x [M] mcd{y,z}
mcd{mcd{x,y},z} =  mcd{x,mcd{y,z}}


mcm{mcm{x,y},z} =  mcm{x,mcm{y,z}}
mcm{mcm{x,y},z} = mcm{x,y} [W] z
mcm{mcm{x,y},z} = (x [W] y) [W] z
mcm{mcm{x,y},z} = x [W] (y [W] z)
mcm{mcm{x,y},z} = x [W] mcm{y,z}
mcm{mcm{x,y},z} =  mcm{x,mcm{y,z}}


mcd{x,y} =  mcd{y,x}
mcm{x,y} =  mcm{y,x}


mcd{x,x} =  x
mcm{x,x} =  x


mcm{mcd{x,y},z}=mcd{mcm{x,z},mcm{y,z}}
mcd{mcm{x,y},z}=mcm{mcd{x,z},mcd{y,z}}


m·k=n <==> m | n <==> mcd{m,n}=m <==> mcm{m,n}=n
(1/n)·k=(1/m) <==> (1/n) | (1/m) <==> mcm{(1/m),(1/n)}=(1/m) <==> mcd{(1/m),(1/n)}=(1/n)

topologia del maxim comú divisor

E_{j} = k^{j}
¬E_{j} = (1/k^{j})


mcd{k^{m},...,k^{n}}=k^{m} & [Aj][ k^{m}·k^{j+(-m)} = k^{j} ]
mcm{k^{m},...,k^{n}}=k^{n} & [Aj][ k^{j}·k^{n+(-j)} = k^{n} ]


mcd{(1/k^{m}),...,(1/k^{n})}=(1/k^{n}) & [Aj][ (1/k^{n})·k^{n+(-j)} = (1/k^{j}) ]
mcm{(1/k^{m}),...,(1/k^{n})}=(1/k^{m}) & [Aj][ (1/k^{j})·k^{j+(-m)}=(1/k^{m}) ]

topologia de singletons de dimensió n

E_{j} = {a_{1},...,a_{j}} = { x : ( x = a_{1} ) or ... or ( x = a_{j} ) }
¬E_{j} = }a_{1},...,a_{j}{ = { x : ¬( x = a_{1} ) & ... & ¬( x = a_{j} ) }


{a_{1},...,a_{m}} [M] ... [M] {a_{1},...,a_{n}} = {a_{1},...,a_{m}}
{a_{1},...,a_{m}} [W] ... [W] {a_{1},...,a_{n}} = {a_{1},...,a_{n}}


}a_{1},...,a_{m}{ [M] ... [M] }a_{1},...,a_{n}{ = }a_{1},...,a_{n}{
}a_{1},...,a_{m}{ [W] ... [W] }a_{1},...,a_{n}{ = }a_{1},...,a_{m}{

topologia de intervals totalment ordenats

E_{j} = [(-j),j]_{K}
¬E_{j} = ¬[(-j),j]_{K}


F_{j} = ((-j),j)_{K}
¬F_{j} = ¬((-j),j)_{K}


Si m [< n ==> [(-m),m]_{K} [M] ... [M] [(-n),n]_{K} = [(-m),m]_{K}
Si m [< n ==> [(-m),m]_{K} [W] ... [W] [(-n),n]_{K} = [(-n),n]_{K}


Si m [< n ==> ¬[(-m),m]_{K} [M] ... [M] ¬[(-n),n]_{K} = ¬[(-n),n]_{K}
Si m [< n ==> ¬[(-m),m]_{K} [W] ... [W] ¬[(-n),n]_{K} = ¬[(-m),m]_{K}


Si m [< n ==> ((-m),m)_{K} [M] ... [M] ((-n),n)_{K} = ((-m),m)_{K}
Si m [< n ==> ((-m),m)_{K} [W] ... [W] ((-n),n)_{K} = ((-n),n)_{K}


Si m [< n ==> ¬((-m),m)_{K} [M] ... [M] ¬((-n),n)_{K} = ¬((-n),n)_{K}
Si m [< n ==> ¬((-m),m)_{K} [W] ... [W] ¬((-n),n)_{K} = ¬((-m),m)_{K}


(-1)+...(n)...+(-1) [< (-1)+...(m)...+(-1) [< x [< 1+...(m)...+1 [< 1+...(n)...+1
x < (-1)+...(n)...+(-1) [< (-1)+...(m)...+(-1) or 1+...(m)...+1 [< 1+...(n)...+1 < x


(-1)+...(n)...+(-1) [< (-1)+...(m)...+(-1) < x < 1+...(m)...+1 [< 1+...(n)...+1
x [< (-1)+...(n)...+(-1) [< (-1)+...(m)...+(-1) or 1+...(m)...+1 [< 1+...(n)...+1 [< x




E_{j} = [0,j]_{K}
¬E_{j} = ¬[0,j]_{K}


F_{j} = (0,j)_{K}
¬F_{j} = ¬(0,j)_{K}


E_{j} = [(-j),0]_{K}
¬E_{j} = ¬[(-j),0]_{K}


F_{j} = ((-j),0)_{K}
¬F_{j} = ¬((-j),0)_{K}

funció símbol de parabola

put-pixel( color[i][j] , 0 , 0 );
put-pixel( color[i][j] , not(not(0)) , not(not(0) ):


for( x=1 ; x [< m ; x++ )
{
put-pixel( color[i][j] , x , not( (x+not(m))·(x+not(m))+m·m );
}


for( x=not(1) ; x >] not(m) ; x-- )
{
put-pixel( color[i][j] , x , (x+m)·(x+m)+not(m·m) );
}

función de los discipulos segun sant Jûan l'stronikiano basado en el evangelio de Mateo

Afirmación:
Vosotros sois la sal de la tierra.
Si la sal se desvirtúa,
con que se salará?
Para nada vale la sal ya,
solo para tirarla a la calle,
y que la gente la pise.


Negación:
Vosotras sois el azúcar de la tierra.
Si el azúcar se desvirtúa,
con que se endulzará?
Para nada vale el azucar ya,
solo para tirarlo a la calle,
y que la gente lo pise.

intervals totalment ordenats

[a,b]_{K} = { x€K : a [< x [< b}
(a,b)_{K} = { x€K : a < x < b}


¬[a,b]_{K} = { x€K : x < a or b < x }
¬(a,b)_{K} = { x€K : x [< a or b [< x }


[a,b]_{K} [M] ¬[a,b]_{K} = { x€K : 0 }
[a,b]_{K} [W] ¬[a,b]_{K} = { x€K : 1 }


(a,b)_{K} [M] ¬(a,b)_{K} = { x€K : 0 }
(a,b)_{K} [W] ¬(a,b)_{K} = { x€K : 1 }


    [a,b]_{K} [M] [c,d]_{K} = [a,b]_{K} <==> ...
... [a,b]_{K} [W] [c,d]_{K} = [c,d]_{K} <==> ...
... [a,b]_{K} [<< [c,d]_{K}


    (a,b)_{K} [M] (c,d)_{K} = (a,b)_{K} <==> ...
... (a,b)_{K} [W] (c,d)_{K} = (c,d)_{K} <==> ...
... (a,b)_{K} [<< (c,d)_{K}


    ¬[a,b]_{K} [W] ¬[c,d]_{K} = ¬[a,b]_{K} <==> ...
... ¬[a,b]_{K} [M] ¬[c,d]_{K} = ¬[c,d]_{K} <==> ...
... ¬[c,d]_{K} [<< ¬[a,b]_{K}


    ¬(a,b)_{K} [W] ¬(c,d)_{K} = ¬(a,b)_{K} <==> ...
... ¬(a,b)_{K} [M] ¬(c,d)_{K} = ¬(c,d)_{K} <==> ...
... ¬(c,d)_{K} [<< ¬(a,b)_{K}