:: Properties of Fuzzy Relation
:: by Noboru Endou , Takashi Mitsuishi and Keiji Ohkubo
::
:: Received June 25, 2001
:: Copyright (c) 2001 Association of Mizar Users

begin

registration

let C1 be non empty set ;
let F be Membership_Func of C1;
cluster proj2 F -> non empty ;
coherence
not rng F is empty

proof end;

end;

theorem Th1: :: FUZZY_4:1

for C1 being non empty set
for F being Membership_Func of C1 holds
( rng F is bounded & ( for x being set st x in dom F holds
F . x <= sup (rng F) ) & ( for x being set st x in dom F holds
F . x >= inf (rng F) ) )

proof end;

theorem :: FUZZY_4:2

for C1 being non empty set
for F, G being Membership_Func of C1 st ( for x being set st x in C1 holds
F . x <= G . x ) holds
sup (rng F) <= sup (rng G)

proof end;

theorem Th3: :: FUZZY_4:3

for C1, C2 being non empty set
for f being RMembership_Func of C1,C2
for c being set holds
( 0 <= f . c & f . c <= 1 )

proof end;

definition

let C1, C2 be non empty set ;
let f be RMembership_Func of C1,C2;
let x, y be set ;
:: original: .
redefine func f . x,y -> Element of REAL ;
coherence
f . x,y is Element of REAL

proof end;

end;

theorem :: FUZZY_4:4

for C1, C2 being non empty set
for f being RMembership_Func of C1,C2
for x, y being set holds
( 0 <= f . x,y & f . x,y <= 1 ) by Th3;

begin

notation

let C1, C2 be non empty set ;
let h be RMembership_Func of C2,C1;
synonym converse h for ~ C1;

end;

definition

let C1, C2 be non empty set ;
let h be RMembership_Func of C2,C1;
:: original: converse
redefine func converse h -> RMembership_Func of C1,C2 means :Def1: :: FUZZY_4:def 1
for x, y being set st [x,y] in [:C1,C2:] holds
it . x,y = h . y,x;
coherence
converse is RMembership_Func of C1,C2

proof end;

compatibility
for b₁ being RMembership_Func of C1,C2 holds
( b₁ = converse iff for x, y being set st [x,y] in [:C1,C2:] holds
b₁ . x,y = h . y,x )

proof end;

end;

:: deftheorem Def1 defines converse FUZZY_4:def 1 :

for C1, C2 being non empty set
for h being RMembership_Func of C2,C1
for b₄ being RMembership_Func of C1,C2 holds
( b₄ = converse h iff for x, y being set st [x,y] in [:C1,C2:] holds
b₄ . x,y = h . y,x );

theorem :: FUZZY_4:5

for C1, C2 being non empty set
for f being RMembership_Func of C1,C2 holds converse (converse f) = f

proof end;

theorem Th6: :: FUZZY_4:6

for C1, C2 being non empty set
for f being RMembership_Func of C1,C2 holds 1_minus (converse f) = converse (1_minus f)

proof end;

theorem Th7: :: FUZZY_4:7

for C1, C2 being non empty set
for f, g being RMembership_Func of C1,C2 holds converse (max f,g) = max (converse f),(converse g)

proof end;

theorem Th8: :: FUZZY_4:8

for C1, C2 being non empty set
for f, g being RMembership_Func of C1,C2 holds converse (min f,g) = min (converse f),(converse g)

proof end;

theorem Th9: :: FUZZY_4:9

for C1, C2 being non empty set
for f, g being RMembership_Func of C1,C2
for x, y being set st x in C1 & y in C2 & f . [x,y] <= g . [x,y] holds
(converse f) . [y,x] <= (converse g) . [y,x]

proof end;

theorem :: FUZZY_4:10

for C1, C2 being non empty set
for f, g being RMembership_Func of C1,C2 st g c= holds
converse g c=

proof end;

theorem :: FUZZY_4:11

for C1, C2 being non empty set
for f, g being RMembership_Func of C1,C2 holds converse (f \ g) = (converse f) \ (converse g)

proof end;

theorem :: FUZZY_4:12

for C1, C2 being non empty set
for f, g being RMembership_Func of C1,C2 holds converse (f \+\ g) = (converse f) \+\ (converse g)

proof end;

begin

definition

let C1, C2, C3 be non empty set ;
let h be RMembership_Func of C1,C2;
let g be RMembership_Func of C2,C3;
let x, z be set ;
assume that
A1: x in C1 and
A2: z in C3 ;
func min h,g,x,z -> Membership_Func of C2 means :Def2: :: FUZZY_4:def 2
for y being Element of C2 holds it . y = min (h . [x,y]),(g . [y,z]);
existence
ex b₁ being Membership_Func of C2 st
for y being Element of C2 holds b₁ . y = min (h . [x,y]),(g . [y,z])

proof end;

uniqueness
for b₁, b₂ being Membership_Func of C2 st ( for y being Element of C2 holds b₁ . y = min (h . [x,y]),(g . [y,z]) ) & ( for y being Element of C2 holds b₂ . y = min (h . [x,y]),(g . [y,z]) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines min FUZZY_4:def 2 :

for C1, C2, C3 being non empty set
for h being RMembership_Func of C1,C2
for g being RMembership_Func of C2,C3
for x, z being set st x in C1 & z in C3 holds
for b₈ being Membership_Func of C2 holds
( b₈ = min h,g,x,z iff for y being Element of C2 holds b₈ . y = min (h . [x,y]),(g . [y,z]) );

definition

let C1, C2, C3 be non empty set ;
let h be RMembership_Func of C1,C2;
let g be RMembership_Func of C2,C3;
func h (#) g -> RMembership_Func of C1,C3 means :Def3: :: FUZZY_4:def 3
for x, z being set st [x,z] in [:C1,C3:] holds
it . x,z = sup (rng (min h,g,x,z));
existence
ex b₁ being RMembership_Func of C1,C3 st
for x, z being set st [x,z] in [:C1,C3:] holds
b₁ . x,z = sup (rng (min h,g,x,z))

proof end;

uniqueness
for b₁, b₂ being RMembership_Func of C1,C3 st ( for x, z being set st [x,z] in [:C1,C3:] holds
b₁ . x,z = sup (rng (min h,g,x,z)) ) & ( for x, z being set st [x,z] in [:C1,C3:] holds
b₂ . x,z = sup (rng (min h,g,x,z)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines (#) FUZZY_4:def 3 :

for C1, C2, C3 being non empty set
for h being RMembership_Func of C1,C2
for g being RMembership_Func of C2,C3
for b₆ being RMembership_Func of C1,C3 holds
( b₆ = h (#) g iff for x, z being set st [x,z] in [:C1,C3:] holds
b₆ . x,z = sup (rng (min h,g,x,z)) );

Lm1: for C1, C2, C3 being non empty set
for f being RMembership_Func of C1,C2
for g, h being RMembership_Func of C2,C3
for x, z being set st x in C1 & z in C3 holds
sup (rng (min f,(max g,h),x,z)) = max (sup (rng (min f,g,x,z))),(sup (rng (min f,h,x,z)))

proof end;

theorem :: FUZZY_4:13

for C1, C2, C3 being non empty set
for f being RMembership_Func of C1,C2
for g, h being RMembership_Func of C2,C3 holds f (#) (max g,h) = max (f (#) g),(f (#) h)

proof end;

Lm2: for C1, C2, C3 being non empty set
for f, g being RMembership_Func of C1,C2
for h being RMembership_Func of C2,C3
for x, z being set st x in C1 & z in C3 holds
sup (rng (min (max f,g),h,x,z)) = max (sup (rng (min f,h,x,z))),(sup (rng (min g,h,x,z)))

proof end;

theorem :: FUZZY_4:14

for C1, C2, C3 being non empty set
for f, g being RMembership_Func of C1,C2
for h being RMembership_Func of C2,C3 holds (max f,g) (#) h = max (f (#) h),(g (#) h)

proof end;

Lm3: for C1, C2, C3 being non empty set
for f being RMembership_Func of C1,C2
for g, h being RMembership_Func of C2,C3
for x, z being set st x in C1 & z in C3 holds
sup (rng (min f,(min g,h),x,z)) <= min (sup (rng (min f,g,x,z))),(sup (rng (min f,h,x,z)))

proof end;

theorem :: FUZZY_4:15

for C1, C2, C3 being non empty set
for f being RMembership_Func of C1,C2
for g, h being RMembership_Func of C2,C3 holds min (f (#) g),(f (#) h) c=

proof end;

Lm4: for C1, C2, C3 being non empty set
for f, g being RMembership_Func of C1,C2
for h being RMembership_Func of C2,C3
for x, z being set st x in C1 & z in C3 holds
sup (rng (min (min f,g),h,x,z)) <= min (sup (rng (min f,h,x,z))),(sup (rng (min g,h,x,z)))

proof end;

theorem :: FUZZY_4:16

for C1, C2, C3 being non empty set
for f, g being RMembership_Func of C1,C2
for h being RMembership_Func of C2,C3 holds min (f (#) h),(g (#) h) c=

proof end;

Lm5: for C1, C2, C3 being non empty set
for f being RMembership_Func of C1,C2
for g being RMembership_Func of C2,C3
for x, z being set st x in C1 & z in C3 holds
sup (rng (min (converse g),(converse f),z,x)) = sup (rng (min f,g,x,z))

proof end;

theorem :: FUZZY_4:17

for C1, C2, C3 being non empty set
for f being RMembership_Func of C1,C2
for g being RMembership_Func of C2,C3 holds converse (f (#) g) = (converse g) (#) (converse f)

proof end;

theorem Th18: :: FUZZY_4:18

for C1, C2, C3 being non empty set
for f, g being RMembership_Func of C1,C2
for h, k being RMembership_Func of C2,C3
for x, z being set st x in C1 & z in C3 & ( for y being set st y in C2 holds
( f . [x,y] <= g . [x,y] & h . [y,z] <= k . [y,z] ) ) holds
(f (#) h) . [x,z] <= (g (#) k) . [x,z]

proof end;

theorem :: FUZZY_4:19

for C1, C2, C3 being non empty set
for f, g being RMembership_Func of C1,C2
for h, k being RMembership_Func of C2,C3 st g c= & k c= holds
g (#) k c=

proof end;

begin

definition

let C1, C2 be non empty set ;
func Imf C1,C2 -> RMembership_Func of C1,C2 means :Def4: :: FUZZY_4:def 4
for x, y being set st [x,y] in [:C1,C2:] holds
( ( x = y implies it . x,y = 1 ) & ( x <> y implies it . x,y = 0 ) );
existence
ex b₁ being RMembership_Func of C1,C2 st
for x, y being set st [x,y] in [:C1,C2:] holds
( ( x = y implies b₁ . x,y = 1 ) & ( x <> y implies b₁ . x,y = 0 ) )

proof end;

uniqueness
for b₁, b₂ being RMembership_Func of C1,C2 st ( for x, y being set st [x,y] in [:C1,C2:] holds
( ( x = y implies b₁ . x,y = 1 ) & ( x <> y implies b₁ . x,y = 0 ) ) ) & ( for x, y being set st [x,y] in [:C1,C2:] holds
( ( x = y implies b₂ . x,y = 1 ) & ( x <> y implies b₂ . x,y = 0 ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines Imf FUZZY_4:def 4 :

for C1, C2 being non empty set
for b₃ being RMembership_Func of C1,C2 holds
( b₃ = Imf C1,C2 iff for x, y being set st [x,y] in [:C1,C2:] holds
( ( x = y implies b₃ . x,y = 1 ) & ( x <> y implies b₃ . x,y = 0 ) ) );

theorem :: FUZZY_4:20

for C1, C2 being non empty set
for c being Element of [:C1,C2:] holds
( (Zmf C1,C2) . c = 0 & (Umf C1,C2) . c = 1 ) by FUNCT_3:def 3;

theorem :: FUZZY_4:21

for C1, C2 being non empty set
for x, y being set st [x,y] in [:C1,C2:] holds
( (Zmf C1,C2) . [x,y] = 0 & (Umf C1,C2) . [x,y] = 1 ) by FUNCT_3:def 3;

Lm6: for C2, C3, C1 being non empty set
for f being RMembership_Func of C2,C3
for x, z being set st x in C1 & z in C3 holds
sup (rng (min (Zmf C1,C2),f,x,z)) = (Zmf C1,C3) . [x,z]

proof end;

theorem Th22: :: FUZZY_4:22

for C2, C3, C1 being non empty set
for f being RMembership_Func of C2,C3 holds (Zmf C1,C2) (#) f = Zmf C1,C3

proof end;

Lm7: for C1, C2, C3 being non empty set
for f being RMembership_Func of C1,C2
for x, z being set st x in C1 & z in C3 holds
sup (rng (min f,(Zmf C2,C3),x,z)) = (Zmf C1,C3) . [x,z]

proof end;

theorem Th23: :: FUZZY_4:23

for C1, C2, C3 being non empty set
for f being RMembership_Func of C1,C2 holds f (#) (Zmf C2,C3) = Zmf C1,C3

proof end;

theorem :: FUZZY_4:24

for C1 being non empty set
for f being RMembership_Func of C1,C1 holds f (#) (Zmf C1,C1) = (Zmf C1,C1) (#) f

proof end;

begin

theorem :: FUZZY_4:25

for X, Y being non empty set
for x being Element of X
for y being Element of Y holds
( ( x = y implies (Imf X,Y) . x,y = 1 ) & ( x <> y implies (Imf X,Y) . x,y = 0 ) )

proof end;

theorem :: FUZZY_4:26

for X, Y being non empty set
for x being Element of X
for y being Element of Y
for f being RMembership_Func of X,Y holds (converse f) . y,x = f . x,y

proof end;