destructor matemàtic expansiva-contractiva y intervals

teorema:

Si x [< y = z ==> x [< z

demostració:

x [< y [< z & z [< y

teorema:

Si x = y [< z ==> x [< z

demostració:

y [< x & x [< y [< z

teorema:

Si x [< y < z ==> x < z

demostració:

x [< y [< z & Absurd: Si x = z ==> ( z = x [< y [< z & y != z )

teorema:

Si x < y [< z ==> x < z

demostració:

x [< y [< z & Absurd: Si x = z ==> (  x [< y [< z = x & x != y )

teorema:

Si x != y = z ==> x != z

demostració:

Absurd: Si x = z ==> y = z = x & x != y

teorema:

Si x = y != z ==> x != z

demostració:

Absurd: Si x = z ==> z = x = y & y != z

teorema:

Si x < y = z ==> x < z

demostració:

x [< y [< z & z [< y

x != y = z

teorema:

Si x = y < z ==> x < z

demostració:

y [< x & x [< y [< z

x = y != z

definició:

[Ax][ x€A ==> min(A) [< x [< max(A) ]

[Ax][ x€A ==> inf(A) < x < sup(A) ]

definició:

[a,b]_{A} = { x€A : a [< x [< b }

(a,b)_{A} = { x€A : a < x < b }

[a,b)_{A} = { x€A : a [< x < b }

(a,b]_{A} = { x€A : a < x [< b }

definició:

a |=| b

101

100

001

000

teorema:

Si R(u,x) |=| R(x,v) ==> R(u,x) & R(x,v).

( ( R(u,x) & R(x,y) ) |=| ( R(y,x) & R(x,v) ) ) <==> R(u,x) |=| R(x,v).

( ( R(u,x) & R(x,y) ) |=| ( R(y,z) & R(z,v) ) ) <==> ...

... ( ( R(u,x) & R(x,y) ) |=| ( R(y,x) & R(x,v) ) ).

teorema constructor:

(a,c]_{A} [ |=| ] [c,b)_{A} [<< (a,b)_{A}

teorema destructor:

Sigui A totalment ordenat.

(a,c]_{A} [ |=| ] [c,b)_{A} [<< [b,a]_{A}

a < x [< c |=| c [< y < b

a < x [< c |=| c [< x < b

a < x |=| x < b

a < x < b

¬( a < x || x < b )

b [< x [< a

teorema constructor:

[a,c)_{A} [ |=| ] (c,b]_{A} [<< [a,b]_{A}

teorema destructor:

Sigui A totalment ordenat.

[a,c)_{A} [ |=| ] (c,b]_{A} [<< (b,a)_{A}

a [< x < c |=| c < y [< b

a [< x < c |=| c < x [< b

a [< x |=| x [< b

a [< x [< b

¬( a [< x || x [< b )

b < x < a

(a,c]_{A} [ |=| ] [c,d]_{A} [ |=| ] [d,b)_{A} [<< (a,b)_{A}

a < x [< c |=| c [< y [< d

a < x [< c |=| c [< x [< d

a < x |=| x [< d

a < x [< d

a < x [< d |=| d [< z < b

a < x [< d |=| d [< x < b

a < x |=| x < b

a < x < b

[a,c)_{A} [ |=| ] (c,d)_{A} [ |=| ] (d,b]_{A} [<< [a,b]_{A}

a [< x < c |=| c < y < d

a [< x < c |=| c < x < d

a [< x |=| x < d

a [< x < d

a [< x < d |=| d < z [< b

a [< x < d |=| d < x [< b

a [< x |=| x [< b

a [< x [< b

(a,c]_{A} [ |=| ] [c,s)_{A} [ |=| ] (s,d]_{A} [ |=| ] [d,b)_{A} [<< (a,b)_{A}

a < x [< c |=| c [< y < s

a < x [< c |=| c [< x < s

a < x |=| x < s

a < x < s

a < x < s |=| s < z [< d

a < x < s |=| s < x [< d

a < x |=| x [< d

a < x [< d

a < x [< d |=| d [< t < b

a < x [< d |=| d [< x < b

a < x |=| x < b

a < x < b

[a,c)_{A} [ |=| ] (c,s]_{A} [ |=| ] [s,d)_{A} [ |=| ] (d,b]_{A} [<< [a,b]_{A}

a [< x < c |=| c < y [< s

a [< x < c |=| c < x [< s

a [< x |=| x [< s

a [< x [< s

a [< x [< s |=| s [< z < d

a [< x [< s |=| s [< x < d

a [< x |=| x < d

a [< x < d

a [< x < d |=| d < t [< b

a [< x < d |=| d < x [< b

a [< x |=| x [< b

a [< x [< b

laboratori de problemes:

Sigui A totalment ordenat.

(a,c]_{A} [ |=| ] [c,s]_{A} [ |=| ] [s,d]_{A} [ |=| ] [d,b)_{A} [<< [b,a]_{A}

[a,c)_{A} [ |=| ] (c,s)_{A} [ |=| ] (s,d)_{A} [ |=| ] (d,b]_{A} [<< (b,a)_{A}

Sigui A,B [<< E & E totalment ordenat.

Si < f: A---> B & x -->f(x) > & max(A) [< min(B) ==> f(x) < x

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

Si < f: A---> B & x -->f(x) > & max(A) < min(B) ==> f(x) [< x

Si < f: A---> B & x -->f(x) > & min(A) >] max(B) ==> f(x) > x

Si < f: A---> B & x -->f(x) > & min(A) > max(B) ==> f(x) >] x

Si < f: [1,n]_{N}---> [n,m]_{N} & x -->f(x) > ==> f(x) < x

1 [< x [< n [< f(x) [< m

Si < f: [(-n),(-1)]_{N} ---> [(-m),(-n)]_{N}  & x -->f(x) > ==> f(x) > x

(-m) [< f(x) [< (-n) [< x [< (-1)

laboratori de problemes:

Si < f: [1,n]_{N}---> [n+1,2n]_{N} & x -->f(x) > ==> f(x) [< x

Si < f: [(-n),(-1)]_{N} ---> [(-2)·n,(-n)+(-1)]_{N}  & x -->f(x) > ==> f(x) >] x

Sigui A totalment ordenat.

Si < f: A ---> A & x --> f(x) = max{ z€A : x [< z } > ==> f(x) < x

x [< z [< max{ z€A : x [< z } = f(x)

Si < f: A ---> A & x --> f(x) = max{ z€A : x < z } > ==> f(x) [< x

Si < f: A ---> A & x --> f(x) = min{ z€A : x >] z } > ==> f(x) > x

Si < f: A ---> A & x --> f(x) = min{ z€A : x > z } > ==> f(x) >] x

Sigui A totalment ordenat.

Si < f: A ---> A & x --> f(x) = sup{ z€A : x [< z } > ==> f(x) [< x

x [< z < sup{ z€A : x [< z } = f(x)

Si < f: A ---> A & x --> f(x) = sup{ z€A : x < z } > ==> f(x) [< x

Si < f: A ---> A & x --> f(x) = inf{ z€A : x >] z } > ==> f(x) >] x

Si < f: A ---> A & x --> f(x) = inf{ z€A : x > z } > ==> f(x) >] x

Sigui A totalment ordenat.

Si < f: A ---> A & x --> f_{n}(x) = max{ z€A : x [< min{z,n} } > ==> f_{n}(x) < x

x [< min{z,n} [< z [< max{ z€A : x [< min{z,n} } = f_{n}(x)

Si < f: A ---> A & x --> f_{n}(x) = max{ z€A : x < min{z,n} } > ==> f_{n}(x) [< x

Si < f: A ---> A & x --> f_{n}(x) = min{ z€A : x >] max{z,(-n)} } > ==> f_{n}(x) > x

Si < f: A ---> A & x --> f_{n}(x) = min{ z€A : x > max{z,(-n)} } > ==> f_{n}(x) >] x

Sigui A totalment ordenat.

Si < f: A ---> A & x --> f_{n}(x) = sup{ z€A : x [< inf{z,n} } > ==> f_{n}(x) [< x

x [< inf{z,n} < z < sup{ z€A : x [< inf{z,n} } = f_{n}(x)

Si < f: A ---> A & x --> f_{n}(x) = sup{ z€A : x < inf{z,n} } > ==> f_{n}(x) [< x

Si < f: A ---> A & x --> f_{n}(x) = inf{ z€A : x >] sup{z,(-n)} } > ==> f_{n}(x) >] x

Si < f: A ---> A & x --> f_{n}(x) = inf{ z€A : x > sup{z,(-n)} } > ==> f_{n}(x) >] x

Sigui A totalment ordenat.

Si < f: A ---> A & x --> f_{n}(x) = max{ z€A : x [< inf{z,n} } > ==> f_{n}(x) [< x

x [< inf{z,n} < z [< max{ z€A : x [< inf{z,n} } = f_{n}(x)

Si < f: A ---> A & x --> f_{n}(x) = max{ z€A : x < inf{z,n} } > ==> f_{n}(x) [< x

Si < f: A ---> A & x --> f_{n}(x) = min{ z€A : x >] sup{z,(-n)} } > ==> f_{n}(x) >] x

Si < f: A ---> A & x --> f_{n}(x) = min{ z€A : x > sup{z,(-n)} } > ==> f_{n}(x) >] x

Sigui A totalment ordenat.

Si < f: A ---> A & x --> f_{n}(x) = sup{ z€A : x [< min{z,n} } > ==> f_{n}(x) [< x

x [< min{z,n} [< z < sup{ z€A : x [< min{z,n} } = f_{n}(x)

Si < f: A ---> A & x --> f_{n}(x) = sup{ z€A : x < min{z,n} } > ==> f_{n}(x) [< x

Si < f: A ---> A & x --> f_{n}(x) = inf{ z€A : x >] max{z,(-n)} } > ==> f_{n}(x) >] x

Si < f: A ---> A & x --> f_{n}(x) = inf{ z€A : x > max{z,(-n)} } > ==> f_{n}(x) >] x