:: Concept of Fuzzy Set and Membership Function and Basic Properties of Fuzzy Set Operation
:: by Takashi Mitsuishi , Noboru Endou and Yasunari Shidama
::
:: Received May 18, 2000
:: Copyright (c) 2000-2025 Association of Mizar Users


:: Definition of Membership Function and Fuzzy Set
registration
let x, y be set ;
cluster K126(x,y) -> [.0,1.] -valued ;
coherence
chi (x,y) is [.0,1.] -valued
proof end;
end;

registration
let C be non empty set ;
cluster Relation-like C -defined REAL -valued [.0,1.] -valued non empty Function-like total V30(C, REAL ) complex-valued ext-real-valued real-valued for Element of K16(K17(C,REAL));
existence
ex b1 being Function of C,REAL st b1 is [.0,1.] -valued
proof end;
end;

definition
let C be non empty set ;
mode Membership_Func of C is [.0,1.] -valued Function of C,REAL;
end;

theorem :: FUZZY_1:1
for C being non empty set holds chi (C,C) is Membership_Func of C ;

registration
let X be non empty set ;
cluster [.0,1.] -valued Function-like V30(X, REAL ) -> real-valued for Element of K16(K17(X,REAL));
coherence
for b1 being Membership_Func of X holds b1 is real-valued ;
end;

definition
let f, g be real-valued Function;
pred f is_less_than g means :: FUZZY_1:def 1
( dom f c= dom g & ( for x being set st x in dom f holds
f . x <= g . x ) );
reflexivity
for f being real-valued Function holds
( dom f c= dom f & ( for x being set st x in dom f holds
f . x <= f . x ) ) ;
end;

:: deftheorem defines is_less_than FUZZY_1:def 1 :
for f, g being real-valued Function holds
( f is_less_than g iff ( dom f c= dom g & ( for x being set st x in dom f holds
f . x <= g . x ) ) );

notation
let X be non empty set ;
let f, g be Membership_Func of X;
synonym f c= g for X is_less_than f;
end;

definition
let X be non empty set ;
let f, g be Membership_Func of X;
:: original: is_less_than
redefine pred f is_less_than g means :Def2: :: FUZZY_1:def 2
 for x being Element of X holds f . x <= g . x;
compatibility
( f is_less_than g iff for x being Element of X holds f . x <= g . x )
proof end;
end;

:: deftheorem Def2 defines is_less_than FUZZY_1:def 2 :
for X being non empty set
for f, g being Membership_Func of X holds
( f is_less_than g iff for x being Element of X holds f . x <= g . x );

Lm1: for C being non empty set
for f, g being Membership_Func of C st g c= & f c= holds
f = g
proof end;

theorem :: FUZZY_1:2
for C being non empty set
for f, g being Membership_Func of C holds
( f = g iff ( g c= & f c= ) ) by Lm1;

theorem :: FUZZY_1:3
for C being non empty set
for f being Membership_Func of C holds f c= ;

theorem :: FUZZY_1:4
for C being non empty set
for f, h, g being Membership_Func of C st g c= & h c= holds
h c= ;

reconsider jj = 1 as Element of REAL by XREAL_0:def 1;

definition
let C be non empty set ;
let h, g be Membership_Func of C;
func min (h,g) -> Membership_Func of C means :Def3: :: FUZZY_1:def 3
 for c being Element of C holds it . c = min ((h . c),(g . c));
existence
ex b1 being Membership_Func of C st
for c being Element of C holds b1 . c = min ((h . c),(g . c))
proof end;
uniqueness
for b1, b2 being Membership_Func of C st ( for c being Element of C holds b1 . c = min ((h . c),(g . c)) ) & ( for c being Element of C holds b2 . c = min ((h . c),(g . c)) ) holds
b1 = b2
proof end;
idempotence
for h being Membership_Func of C
for c being Element of C holds h . c = min ((h . c),(h . c)) ;
commutativity
for b1, h, g being Membership_Func of C st ( for c being Element of C holds b1 . c = min ((h . c),(g . c)) ) holds
for c being Element of C holds b1 . c = min ((g . c),(h . c)) ;
end;

:: deftheorem Def3 defines min FUZZY_1:def 3 :
for C being non empty set
for h, g, b4 being Membership_Func of C holds
( b4 = min (h,g) iff for c being Element of C holds b4 . c = min ((h . c),(g . c)) );

definition
let C be non empty set ;
let h, g be Membership_Func of C;
func max (h,g) -> Membership_Func of C means :Def4: :: FUZZY_1:def 4
 for c being Element of C holds it . c = max ((h . c),(g . c));
existence
ex b1 being Membership_Func of C st
for c being Element of C holds b1 . c = max ((h . c),(g . c))
proof end;
uniqueness
for b1, b2 being Membership_Func of C st ( for c being Element of C holds b1 . c = max ((h . c),(g . c)) ) & ( for c being Element of C holds b2 . c = max ((h . c),(g . c)) ) holds
b1 = b2
proof end;
idempotence
for h being Membership_Func of C
for c being Element of C holds h . c = max ((h . c),(h . c)) ;
commutativity
for b1, h, g being Membership_Func of C st ( for c being Element of C holds b1 . c = max ((h . c),(g . c)) ) holds
for c being Element of C holds b1 . c = max ((g . c),(h . c)) ;
end;

:: deftheorem Def4 defines max FUZZY_1:def 4 :
for C being non empty set
for h, g, b4 being Membership_Func of C holds
( b4 = max (h,g) iff for c being Element of C holds b4 . c = max ((h . c),(g . c)) );

definition
let C be non empty set ;
let h be Membership_Func of C;
func 1_minus h -> Membership_Func of C means :Def5: :: FUZZY_1:def 5
 for c being Element of C holds it . c = 1 - (h . c);
existence
ex b1 being Membership_Func of C st
for c being Element of C holds b1 . c = 1 - (h . c)
proof end;
uniqueness
for b1, b2 being Membership_Func of C st ( for c being Element of C holds b1 . c = 1 - (h . c) ) & ( for c being Element of C holds b2 . c = 1 - (h . c) ) holds
b1 = b2
proof end;
involutiveness
for b1, b2 being Membership_Func of C st ( for c being Element of C holds b1 . c = 1 - (b2 . c) ) holds
for c being Element of C holds b2 . c = 1 - (b1 . c)
proof end;
end;

:: deftheorem Def5 defines 1_minus FUZZY_1:def 5 :
for C being non empty set
for h, b3 being Membership_Func of C holds
( b3 = 1_minus h iff for c being Element of C holds b3 . c = 1 - (h . c) );

theorem :: FUZZY_1:5
for C being non empty set
for c being Element of C
for h, g being Membership_Func of C holds
( min ((h . c),(g . c)) = (min (h,g)) . c & max ((h . c),(g . c)) = (max (h,g)) . c ) by Def3, Def4;

theorem :: FUZZY_1:6
for C being non empty set
for f, h, g being Membership_Func of C holds
( max (h,h) = h & min (h,h) = h & max (h,h) = min (h,h) & min (f,g) = min (g,f) & max (f,g) = max (g,f) ) ;

theorem Th7: :: FUZZY_1:7
for C being non empty set
for f, h, g being Membership_Func of C holds
( max ((max (f,g)),h) = max (f,(max (g,h))) & min ((min (f,g)),h) = min (f,(min (g,h))) )
proof end;

theorem Th8: :: FUZZY_1:8
for C being non empty set
for f, g being Membership_Func of C holds
( max (f,(min (f,g))) = f & min (f,(max (f,g))) = f )
proof end;

theorem Th9: :: FUZZY_1:9
for C being non empty set
for f, h, g being Membership_Func of C holds
( min (f,(max (g,h))) = max ((min (f,g)),(min (f,h))) & max (f,(min (g,h))) = min ((max (f,g)),(max (f,h))) )
proof end;

theorem :: FUZZY_1:10
canceled;

::$CT
theorem Th10: :: FUZZY_1:11
for C being non empty set
for f, g being Membership_Func of C holds
( 1_minus (max (f,g)) = min ((1_minus f),(1_minus g)) & 1_minus (min (f,g)) = max ((1_minus f),(1_minus g)) )
proof end;

:: Empty Fuzzy Set and Universal Fuzzy Set
theorem :: FUZZY_1:12
for C being non empty set holds chi ({},C) is Membership_Func of C ;

definition
let C be non empty set ;
func EMF C -> Membership_Func of C equals :: FUZZY_1:def 6
 chi ({},C);
correctness
coherence
chi ({},C) is Membership_Func of C;
;
end;

:: deftheorem defines EMF FUZZY_1:def 6 :
for C being non empty set holds EMF C = chi ({},C);

definition
let C be non empty set ;
func UMF C -> Membership_Func of C equals :: FUZZY_1:def 7
 chi (C,C);
correctness
coherence
chi (C,C) is Membership_Func of C;
;
end;

:: deftheorem defines UMF FUZZY_1:def 7 :
for C being non empty set holds UMF C = chi (C,C);

theorem Th12: :: FUZZY_1:13
for C being non empty set
for a, b being Element of REAL
for f being PartFunc of C,REAL st rng f c= [.a,b.] & a <= b holds
for x being Element of C st x in dom f holds
( a <= f . x & f . x <= b )
proof end;

theorem Th13: :: FUZZY_1:14
for C being non empty set
for f being Membership_Func of C holds f c=
proof end;

theorem Th14: :: FUZZY_1:15
for C being non empty set
for f being Membership_Func of C holds UMF C c=
proof end;

theorem Th15: :: FUZZY_1:16
for C being non empty set
for x being Element of C
for h being Membership_Func of C holds
( (EMF C) . x <= h . x & h . x <= (UMF C) . x ) by Th13, Th14, Def2;

theorem Th16: :: FUZZY_1:17
for C being non empty set
for f, g being Membership_Func of C holds
( f c= & max (f,g) c= )
proof end;

theorem Th17: :: FUZZY_1:18
for C being non empty set
for f being Membership_Func of C holds
( max (f,(UMF C)) = UMF C & min (f,(UMF C)) = f & max (f,(EMF C)) = f & min (f,(EMF C)) = EMF C )
proof end;

theorem Th18: :: FUZZY_1:19
for C being non empty set
for f, h, g being Membership_Func of C st h c= & h c= holds
h c=
proof end;

theorem :: FUZZY_1:20
for C being non empty set
for f, h, g being Membership_Func of C st g c= holds
max (g,h) c=
proof end;

theorem :: FUZZY_1:21
for C being non empty set
for f, h, g, h1 being Membership_Func of C st g c= & h1 c= holds
max (g,h1) c=
proof end;

theorem :: FUZZY_1:22
for C being non empty set
for f, g being Membership_Func of C st g c= holds
max (f,g) = g
proof end;

theorem :: FUZZY_1:23
for C being non empty set
for f, g being Membership_Func of C holds max (f,g) c=
proof end;

theorem Th23: :: FUZZY_1:24
for C being non empty set
for f, h, g being Membership_Func of C st f c= & g c= holds
min (f,g) c=
proof end;

theorem :: FUZZY_1:25
for C being non empty set
for f, h, g being Membership_Func of C st g c= holds
min (g,h) c=
proof end;

theorem :: FUZZY_1:26
for C being non empty set
for f, h, g, h1 being Membership_Func of C st g c= & h1 c= holds
min (g,h1) c=
proof end;

theorem Th26: :: FUZZY_1:27
for C being non empty set
for f, g being Membership_Func of C st g c= holds
min (f,g) = f
proof end;

theorem :: FUZZY_1:28
for C being non empty set
for f, h, g being Membership_Func of C st g c= & h c= & min (g,h) = EMF C holds
f = EMF C
proof end;

theorem :: FUZZY_1:29
for C being non empty set
for f, h, g being Membership_Func of C st max ((min (f,g)),(min (f,h))) = f holds
max (g,h) c=
proof end;

theorem :: FUZZY_1:30
for C being non empty set
for f, h, g being Membership_Func of C st g c= & min (g,h) = EMF C holds
min (f,h) = EMF C
proof end;

theorem :: FUZZY_1:31
for C being non empty set
for f being Membership_Func of C st EMF C c= holds
f = EMF C
proof end;

theorem :: FUZZY_1:32
for C being non empty set
for f, g being Membership_Func of C holds
( max (f,g) = EMF C iff ( f = EMF C & g = EMF C ) )
proof end;

theorem :: FUZZY_1:33
for C being non empty set
for f, h, g being Membership_Func of C holds
( f = max (g,h) iff ( f c= & f c= & ( for h1 being Membership_Func of C st h1 c= & h1 c= holds
h1 c= ) ) )
proof end;

theorem :: FUZZY_1:34
for C being non empty set
for f, h, g being Membership_Func of C holds
( f = min (g,h) iff ( g c= & h c= & ( for h1 being Membership_Func of C st g c= & h c= holds
f c= ) ) )
proof end;

theorem :: FUZZY_1:35
for C being non empty set
for f, h, g being Membership_Func of C st max (g,h) c= & min (f,h) = EMF C holds
g c=
proof end;

Lm2: for C being non empty set
for f, g being Membership_Func of C st g c= holds
1_minus f c=
proof end;

theorem Th35: :: FUZZY_1:36
for C being non empty set
for f, g being Membership_Func of C holds
( g c= iff 1_minus f c= )
proof end;

theorem :: FUZZY_1:37
for C being non empty set
for f, g being Membership_Func of C st 1_minus g c= holds
1_minus f c=
proof end;

theorem :: FUZZY_1:38
for C being non empty set
for f, g being Membership_Func of C holds 1_minus f c= by Th16, Th35;

theorem :: FUZZY_1:39
for C being non empty set
for f, g being Membership_Func of C holds 1_minus (min (f,g)) c= by Th16, Th35;

theorem Th39: :: FUZZY_1:40
for C being non empty set holds
( 1_minus (EMF C) = UMF C & 1_minus (UMF C) = EMF C )
proof end;

:: Exclusive Sum, Absolute Difference
definition
let C be non empty set ;
let h, g be Membership_Func of C;
func h \+\ g -> Membership_Func of C equals :: FUZZY_1:def 8
 max ((min (h,(1_minus g))),(min ((1_minus h),g)));
coherence
max ((min (h,(1_minus g))),(min ((1_minus h),g))) is Membership_Func of C ;
commutativity
for b1, h, g being Membership_Func of C st b1 = max ((min (h,(1_minus g))),(min ((1_minus h),g))) holds
b1 = max ((min (g,(1_minus h))),(min ((1_minus g),h))) ;
end;

:: deftheorem defines \+\ FUZZY_1:def 8 :
for C being non empty set
for h, g being Membership_Func of C holds h \+\ g = max ((min (h,(1_minus g))),(min ((1_minus h),g)));

theorem :: FUZZY_1:41
for C being non empty set
for f being Membership_Func of C holds f \+\ (EMF C) = f
proof end;

theorem :: FUZZY_1:42
for C being non empty set
for f being Membership_Func of C holds f \+\ (UMF C) = 1_minus f
proof end;

theorem :: FUZZY_1:43
for C being non empty set
for f, h, g being Membership_Func of C holds min ((min ((max (f,g)),(max (g,h)))),(max (h,f))) = max ((max ((min (f,g)),(min (g,h)))),(min (h,f)))
proof end;

theorem :: FUZZY_1:44
for C being non empty set
for f, g being Membership_Func of C holds 1_minus (f \+\ g) c=
proof end;

theorem :: FUZZY_1:45
for C being non empty set
for f, g being Membership_Func of C holds max (f,g) c=
proof end;

theorem :: FUZZY_1:46
for C being non empty set
for f being Membership_Func of C holds f \+\ f = min (f,(1_minus f)) ;

definition
let C be non empty set ;
let h, g be Membership_Func of C;
func ab_difMF (h,g) -> Membership_Func of C means :: FUZZY_1:def 9
 for c being Element of C holds it . c = |.((h . c) - (g . c)).|;
existence
ex b1 being Membership_Func of C st
for c being Element of C holds b1 . c = |.((h . c) - (g . c)).|
proof end;
uniqueness
for b1, b2 being Membership_Func of C st ( for c being Element of C holds b1 . c = |.((h . c) - (g . c)).| ) & ( for c being Element of C holds b2 . c = |.((h . c) - (g . c)).| ) holds
b1 = b2
proof end;
end;

:: deftheorem defines ab_difMF FUZZY_1:def 9 :
for C being non empty set
for h, g, b4 being Membership_Func of C holds
( b4 = ab_difMF (h,g) iff for c being Element of C holds b4 . c = |.((h . c) - (g . c)).| );