:: FUZZY_1 semantic presentation
theorem Th1: :: FUZZY_1:1
:: deftheorem Def1 defines Membership_Func FUZZY_1:def 1 :
theorem Th2: :: FUZZY_1:2
:: deftheorem Def2 defines is_less_than FUZZY_1:def 2 :
:: deftheorem Def3 defines is_less_than FUZZY_1:def 3 :
Lemma35:
for C being non empty set
for f, g being Membership_Func of C st g c= & f c= holds
f = g
theorem Th3: :: FUZZY_1:3
theorem Th4: :: FUZZY_1:4
theorem Th5: :: FUZZY_1:5
definition
let C be non
empty set ;
let h be
Membership_Func of
C,
g be
Membership_Func of
C;
func min c2,
c3 -> Membership_Func of
a1 means :
Def4:
:: FUZZY_1:def 4
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)
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
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 Def4 defines min FUZZY_1:def 4 :
definition
let C be non
empty set ;
let h be
Membership_Func of
C,
g be
Membership_Func of
C;
func max c2,
c3 -> Membership_Func of
a1 means :
Def5:
:: FUZZY_1:def 5
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)
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
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 Def5 defines max FUZZY_1:def 5 :
:: deftheorem Def6 defines 1_minus FUZZY_1:def 6 :
theorem Th6: :: FUZZY_1:6
theorem Th7: :: FUZZY_1:7
for
C being non
empty set for
h,
f,
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 Th8: :: FUZZY_1:8
for
C being non
empty set for
f,
g,
h 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) )
theorem Th9: :: FUZZY_1:9
theorem Th10: :: FUZZY_1:10
for
C being non
empty set for
f,
g,
h 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) )
theorem Th11: :: FUZZY_1:11
theorem Th12: :: FUZZY_1:12
theorem Th13: :: FUZZY_1:13
:: deftheorem Def7 defines EMF FUZZY_1:def 7 :
:: deftheorem Def8 defines UMF FUZZY_1:def 8 :
theorem Th14: :: FUZZY_1:14
theorem Th15: :: FUZZY_1:15
theorem Th16: :: FUZZY_1:16
theorem Th17: :: FUZZY_1:17
theorem Th18: :: FUZZY_1:18
theorem Th19: :: FUZZY_1:19
theorem Th20: :: FUZZY_1:20
theorem Th21: :: FUZZY_1:21
theorem Th22: :: FUZZY_1:22
theorem Th23: :: FUZZY_1:23
theorem Th24: :: FUZZY_1:24
theorem Th25: :: FUZZY_1:25
theorem Th26: :: FUZZY_1:26
theorem Th27: :: FUZZY_1:27
theorem Th28: :: FUZZY_1:28
theorem Th29: :: FUZZY_1:29
theorem Th30: :: FUZZY_1:30
theorem Th31: :: FUZZY_1:31
theorem Th32: :: FUZZY_1:32
theorem Th33: :: FUZZY_1:33
theorem Th34: :: FUZZY_1:34
theorem Th35: :: FUZZY_1:35
theorem Th36: :: FUZZY_1:36
theorem Th37: :: FUZZY_1:37
theorem Th38: :: FUZZY_1:38
Lemma74:
for C being non empty set
for f, g being Membership_Func of C st g c= holds
1_minus f c=
theorem Th39: :: FUZZY_1:39
theorem Th40: :: FUZZY_1:40
theorem Th41: :: FUZZY_1:41
theorem Th42: :: FUZZY_1:42
theorem Th43: :: FUZZY_1:43
theorem Th44: :: FUZZY_1:44
definition
let C be non
empty set ;
let h be
Membership_Func of
C,
g be
Membership_Func of
C;
func c2 \+\ c3 -> Membership_Func of
a1 equals :: FUZZY_1:def 9
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 Def9 defines \+\ FUZZY_1:def 9 :
theorem Th45: :: FUZZY_1:45
theorem Th46: :: FUZZY_1:46
theorem Th47: :: FUZZY_1:47
for
C being non
empty set for
f,
g,
h 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)
theorem Th48: :: FUZZY_1:48
theorem Th49: :: FUZZY_1:49
theorem Th50: :: FUZZY_1:50
:: deftheorem Def10 defines ab_difMF FUZZY_1:def 10 :