isomorfismes de ordre

Sigui A un conjunt ordenat & < f: A ---> A & x --> f(x) >

Si f(x) és un isomorfisme de odre ==> ...

... ( f(x) és expansiva <==> f^{o(-1)}(x) és contractiva ).

... ( f(x) és contractiva <==> f^{o(-1)}(x) és expansiva ).

Demostració:

x [< f(x)

f^{o(-1)}(x) [< x

f(x) [< x

x [< f^{o(-1)}(x)

Demostració particular:

0 [< n

x [< x+n

0 >] (-n)

x >] x+(-n)

Demostració particular:

Sigui x >] 0 ==>

1 [< n

x [< nx

1 >] (1/n)

x >] (x/n)

Sigui x [< 0 ==>

1 [< n

x >] nx

1 >] (1/n)

x [< (x/n)

Sigui A un conjunt ordenat amb mínim y màxim & < f: A ---> A & x --> f(x) >

Si f(x) és un isomorfisme de ordre ==> ...

... Si ( f((-1)·min(A)) = max(A) & f((-1)·max(A)) = min(A) ) ==> [Ec][ f(-c) = c ]

Demostració:

a [< x [< b

(-b) [< (-x) [< (-a)

f(-b) [< f(-x) [< f(-a)

a [< f(-x) [< b

|f(-x)+(-x)| [< b+(-a)

Se defienish un x = c & |f(-c)+(-c)| = 0 ==>

f(-c) = c

Demostració particular:

0 [< n [< m

f(-n) = m+(-n)

f(0) = m

f(-m) = 0

f((-1)·(m/2)) = (m/2)

Sigui A un conjunt ordenat amb mínim y màxim & < f: A ---> A & x --> f(x) >

Si f(x) és un isomorfisme de ordre ==> ...

... Si ( f(min(A)) = (-1)·max(A) & f(max(A)) = (-1)·min(A) ) ==> [Ec][ f(c) = (-c) ]

Demostració:

a [< x [< b

f(a) [< f(x) [< f(b)

(-b) [< f(x) [< (-a)

|f(x)+x| [< b+(-a)

Se defienish un x = c & |f(c)+c| = 0 ==>

f(c) = (-c)

Demostració particular:

0 [< n [< m

f(n) = (-m)+n

f(0) = (-m)

f(m) = 0

f(m/2) = (-1)·(m/2)

funció expansiva y contractiva

Si ( a [< b [< c & < f:[a,b]_{N}⟶[b,c]_{N} & x ↦ f(x) > ==> f(x) es expansiva.

a [< x [< b [< f(x) [< c

Si ( a [< b [< c & < f:[b,c]_{N}⟶[a,b]_{N} & x ↦ f(x) > ==> f(x) es contractiva.

a [< f(x) [< b [< x [< c

productes connectius

A [⋀] B = {<x,y> : x€A ⋀ y€B }
A [⋁] B = {<x,y> : x€A ⋁ y€B }


x€A ⋀ x€B <==> x€A[M]B
x€A ⋁ x€B <==> x€A[W]B


A[xor] B = {<x,y> : x€A xor y€B }


x€A xor x€B <==> x€(A[W]B) [-] (A[M]B)


A [⋀] B = {<x,y> : x€{a,b} ⋀ y€{a,c} } = { <a,a>,<a,c>,<b,a>,<b,c> }
A [⋁] B = {<x,y> : x€{a,b} ⋁ y€{a,c} } = { <a,a>,<a,c>,<b,a>,<b,c>,<b,b>,<c,c> }
A [xor] B = {<x,y> : x€{a,b} xor y€{a,c} } = { <b,b>,<c,c> }


x€A ⋀ y€A <==> y€A ⋀ x€A
x€B ⋀ y€B <==> y€B ⋀ x€B

funció álef

ℵ_{0} = 1
ℵ_{1} = oo
ℵ_{2} = oo^{oo}
ℵ_{3} = oo^{oo^{oo}}

potencies y exponencials infinites

k^{oo} = k·oo^{(k+(-1))}
k^{(oo+(-1))} = oo^{(k+(-1))}


(k+1)^{oo} = (k+1)·oo^{k}
(k+1)^{oo+(-1)} = oo^{k}


1^{(oo+(-1))} = oo^{(1+(-1))} = oo^{0} = 1
(0+1)^{(oo+(-1))} = oo^{0} = 1


oo^{oo} = oo·oo^{(oo+(-1))} = oo·( oo^{oo}/oo )
( (oo+(-1))+1 )^{(oo+(-1))} = oo^{(oo+(-1))} = ( oo^{oo}/oo )

índex de matemàtiques

index de etiquetes matemàtiques

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}

intersecció y reunió de conjunts

x = { t : t€x }
¬x = { t : ¬( t€x ) }


x [M] y ={ t : t€x  &  t€y }
x [W] y ={ t : t€x  or  t€y }


¬x [M] ¬y ={ t : ¬( t€x ) & ¬( t€y ) }
¬x [W] ¬y ={ t : ¬( t€x ) or ¬( t€y ) }


( x [M] y ) [M] z = x [M] ( y [M] z )
( x [W] y ) [W] z = x [W] ( y [W] z )


( ¬x [M] ¬y ) [M] ¬z = ¬x [M] ( ¬y [M] ¬z )
( ¬x [W] ¬y ) [W] ¬z = ¬x [W] ( ¬y [W] ¬z )


( x [M] y ) = ( y [M] x )
( x [W] y ) = ( y [W] x )


( ¬x [M] ¬y ) = ( ¬y [M] ¬x )
( ¬x [W] ¬y ) = ( ¬y [W] ¬x )


( x [M] x ) = x
( x [W] x ) = x


( ¬x [M] ¬x ) = ¬x
( ¬x [W] ¬x ) = ¬x


( x [W] y ) [M] z = ( x [M] z ) [W] ( y [M] z )
( x [M] y ) [W] z = ( x [W] z ) [M] ( y [W] z )


( ¬x [W] ¬y ) [M] ¬z = ( ¬x [M] ¬z ) [W] ( ¬y [M] ¬z )
( ¬x [M] ¬y ) [W] ¬z = ( ¬x [W] ¬z ) [M] ( ¬y [W] ¬z )


( x [M] ¬x ) = { t : 0 }
( x [W] ¬x ) = { t : 1 }


¬( x [M] y ) = ( ¬y [W] ¬x )
¬( x [W] y ) = ( ¬y [M] ¬x )


¬( ¬x [M] ¬y ) = ( y [W] x )
¬( ¬x [W] ¬y ) = ( y [M] x )

funcions injectives y bijectives

< g: oo ---> oo^{oo} & n  -->  g(n) = < f_{n}:oo--->oo & k --> f_{n}(k)=n > >


g(n)=g(m)
f_{n}=f_{m}
sigui k€N ==>
f_{n}(k)=f_{m}(k)
n=m
g és injectiva


< g: R-{0} ---> R-{m} & x  -->  g(x) = x+m >


g(x)=g(y)
x+m=y+m
x=y
g és injectiva


< h: R-{m} ---> R-{0} & x  -->  h(x+m) = x >


h(x+m)=h(y+m)
x=y
x+m=y+m
h és injectiva


< g: R-{1} ---> R-{m} & x  -->  g(x) = mx >
< h: R-{m} ---> R-{1} & x  -->  h(x) = (1/m)·x >


< g: R-{n} ---> R-{m} & x  -->  g(x) = (m/n)·x >
< h: R-{m} ---> R-{n} & x  -->  h(x) = (n/m)·x >


< g: R-{1,2} ---> R-{2,7} & x  -->  g(x) = 5x+(-3) >
< h: R-{2,7} ---> R-{1,2} & x  -->  h(x) = (x+3)/5 >

intersecció y reunió de conjunts

x = { t : t€x }
¬x = { t : ¬( t€x ) }


x [M] y ={ t : t€x  &  t€y }
x [W] y ={ t : t€x  or  t€y }


¬x [M] ¬y ={ t : ¬( t€x ) & ¬( t€y ) }
¬x [W] ¬y ={ t : ¬( t€x ) or ¬( t€y ) }


x [M] y [<< x [W] y
¬x [M] ¬y [<< ¬x [W] ¬y


x [M] y [<< x
x [M] y [<< y


¬x [M] ¬y [<< ¬x
¬x [M] ¬y [<< ¬y


x [<< x [W] y
y [<< x [W] y


¬x [<< ¬x [W] ¬y
¬y [<< ¬x [W] ¬y


x [<< y <==> x [M] y = x  <==> x [W] y = y
¬y [<< ¬x <==> ¬x [M] ¬y = ¬y  <==> ¬x [W] ¬y = ¬x

singletons de elements de conjunts

z€{x} <==> z = x
z€}x{ <==> ¬( z = x )


1 = {0}
(-1) = }0{


z€{x,y} <==> ( z = x or z = y )
z€}x,y{ <==> ( ¬( z = x ) & ¬( z = y ) )


2 = {0,{0}}
(-2) = }0,{0}{


{x} [<< {x,y}
}x{ >>] }x,y{


{x} [W] {y} = {x,y}
}x{ [M] }y{ = }x,y{


[M]{x,y} = x [M] y <==> (  t€[M]{x,y} <==> [Az][ z€{x,y} ==> t€z ] )
[W]{x,y} = x [W] y <==> (  t€[W]{x,y} <==> [Ez][ z€{x,y} & t€z ] )


[M]}x,y{ = ¬( x [W] y ) = ( ¬x [M] ¬y ) <==> (  t€[M]}x,y{ <==> [Az][ z€}x,y{ ==> t€z ] )
[W]}x,y{ = ¬( x [M] y ) = ( ¬x [W] ¬y ) <==> (  t€[W]}x,y{ <==> [Ez][ z€}x,y{ & t€z ] )

funcions expansives y contractives

si < f:A---->B [&] x --> f(x) > & max(A) [< min(B)  ==> x [< f(x).


x [< max(A) [< min(B) [< f(x)


si < f:A---->B [&] x --> f(x) > & min(A) >] max(B)  ==> x >] f(x).


x >] min(A) >] max(B) >] f(x)

funcions expansives y contractives

Si f(n)  = min{z: z [< n } ==> f(n) [< n
f(n) [< z
f(n) [< z [< n


Si f(n)  = max{z: z >] n } ==> f(n) >] n
f(n) >] z
f(n) >] z >] n


Si f(n)  = min{z: z+n [< n } & 0 [< n ==> f(n) [< n
f(n) [< z
f(n)+n [< z+n
0 [< n
f(n) [< f(n)+n
f(n) [< f(n)+n [< z+n [< n


Si f(n)  = max{z: z+n >] n } & 0 >] n ==> f(n) >] n
f(n) >] z
f(n)+n >] z+n
0 >] n
f(n) >] f(n)+n
f(n) >] f(n)+n >] z+n >] n


Si f(n)  = min{z: max{z,n} [< n } ==> f(n) [< n
f(n) [< z
max{f(n),n} [< max{z,n}
f(n) [< max{f(n),n} [< max{z,n} [< n


Si f(n)  = max{z: min{z,n} >] n } ==> f(n) >] n
f(n) >] z
min{f(n),n} >] min{z,n}
f(n) >] min{f(n),n} >] min{z,n} >] n