:: Properties of Fuzzy Relation
:: by Noboru Endou , Takashi Mitsuishi and Keiji Ohkubo
::
:: Received June 25, 2001
:: Copyright (c) 2001-2021 Association of Mizar Users

registration

let C1 be non empty set ;
let F be Membership_Func of C1;

cluster rng F -> non empty ;

coherence
not rng F is empty

proof end;

end;

reconsider jj = 1 as Real ;

theorem Th1: :: FUZZY_4:1

for C1 being non empty set
for F being Membership_Func of C1 holds
( rng F is real-bounded & ( for x being set st x in dom F holds
F . x <= upper_bound (rng F) ) & ( for x being set st x in dom F holds
F . x >= lower_bound (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
upper_bound (rng F) <= upper_bound (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;

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;

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 object 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 object 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 object 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;

:: Definition and properties of the composition

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 object ;
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 object 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 object st [x,z] in [:C1,C3:] holds
it . (x,z) = upper_bound (rng (min (h,g,x,z)));

existence
ex b₁ being RMembership_Func of C1,C3 st
for x, z being object st [x,z] in [:C1,C3:] holds
b₁ . (x,z) = upper_bound (rng (min (h,g,x,z)))

proof end;

uniqueness
for b₁, b₂ being RMembership_Func of C1,C3 st ( for x, z being object st [x,z] in [:C1,C3:] holds
b₁ . (x,z) = upper_bound (rng (min (h,g,x,z))) ) & ( for x, z being object st [x,z] in [:C1,C3:] holds
b₂ . (x,z) = upper_bound (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 object st [x,z] in [:C1,C3:] holds
b₆ . (x,z) = upper_bound (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
upper_bound (rng (min (f,(max (g,h)),x,z))) = max ((upper_bound (rng (min (f,g,x,z)))),(upper_bound (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
upper_bound (rng (min ((max (f,g)),h,x,z))) = max ((upper_bound (rng (min (f,h,x,z)))),(upper_bound (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
upper_bound (rng (min (f,(min (g,h)),x,z))) <= min ((upper_bound (rng (min (f,g,x,z)))),(upper_bound (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
upper_bound (rng (min ((min (f,g)),h,x,z))) <= min ((upper_bound (rng (min (f,h,x,z)))),(upper_bound (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
upper_bound (rng (min ((converse g),(converse f),z,x))) = upper_bound (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;

Lm6: ( 1 in REAL & 0 in REAL )
by XREAL_0:def 1;

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 object 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 object 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 object 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 object 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 object 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;

Lm7: for C1, C2, C3 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
upper_bound (rng (min ((Zmf (C1,C2)),f,x,z))) = (Zmf (C1,C3)) . [x,z]

proof end;

theorem Th22: :: FUZZY_4:22

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

proof end;

Lm8: 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
upper_bound (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;

:: missing, 2006.12.03, AT

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) by Def1, ZFMISC_1:87;