:: 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) 2000-2021 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)).| );