:: FUZZY_4 semantic presentation
theorem Th1: :: FUZZY_4:1
theorem Th2: :: FUZZY_4:2
theorem Th3: :: FUZZY_4:3
theorem Th4: :: FUZZY_4:4
definition
let c
1, c
2 be non
empty set ;
let c
3 be
RMembership_Func of c
2,c
1;
func converse c
3 -> RMembership_Func of a
1,a
2 means :
Def1:
:: FUZZY_4:def 1
for b
1, b
2 being
set holds
(
[b1,b2] in [:a1,a2:] implies a
4 . [b1,b2] = a
3 . [b2,b1] );
existence
ex b1 being RMembership_Func of c1,c2 st
for b2, b3 being set holds
( [b2,b3] in [:c1,c2:] implies b1 . [b2,b3] = c3 . [b3,b2] )
uniqueness
for b1, b2 being RMembership_Func of c1,c2 holds
( ( for b3, b4 being set holds
( [b3,b4] in [:c1,c2:] implies b1 . [b3,b4] = c3 . [b4,b3] ) ) & ( for b3, b4 being set holds
( [b3,b4] in [:c1,c2:] implies b2 . [b3,b4] = c3 . [b4,b3] ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines converse FUZZY_4:def 1 :
theorem Th5: :: FUZZY_4:5
theorem Th6: :: FUZZY_4:6
theorem Th7: :: FUZZY_4:7
theorem Th8: :: FUZZY_4:8
theorem Th9: :: FUZZY_4:9
theorem Th10: :: FUZZY_4:10
theorem Th11: :: FUZZY_4:11
theorem Th12: :: FUZZY_4:12
definition
let c
1, c
2, c
3 be non
empty set ;
let c
4 be
RMembership_Func of c
1,c
2;
let c
5 be
RMembership_Func of c
2,c
3;
let c
6, c
7 be
set ;
assume E8:
( c
6 in c
1 & c
7 in c
3 )
;
func min c
4,c
5,c
6,c
7 -> Membership_Func of a
2 means :
Def2:
:: FUZZY_4:def 2
for b
1 being
Element of a
2 holds a
8 . b
1 = min (a4 . [a6,b1]),
(a5 . [b1,a7]);
existence
ex b1 being Membership_Func of c2 st
for b2 being Element of c2 holds b1 . b2 = min (c4 . [c6,b2]),(c5 . [b2,c7])
uniqueness
for b1, b2 being Membership_Func of c2 holds
( ( for b3 being Element of c2 holds b1 . b3 = min (c4 . [c6,b3]),(c5 . [b3,c7]) ) & ( for b3 being Element of c2 holds b2 . b3 = min (c4 . [c6,b3]),(c5 . [b3,c7]) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines min FUZZY_4:def 2 :
for b
1, b
2, b
3 being non
empty set for b
4 being
RMembership_Func of b
1,b
2for b
5 being
RMembership_Func of b
2,b
3for b
6, b
7 being
set holds
( b
6 in b
1 & b
7 in b
3 implies for b
8 being
Membership_Func of b
2 holds
( b
8 = min b
4,b
5,b
6,b
7 iff for b
9 being
Element of b
2 holds b
8 . b
9 = min (b4 . [b6,b9]),
(b5 . [b9,b7]) ) );
definition
let c
1, c
2, c
3 be non
empty set ;
let c
4 be
RMembership_Func of c
1,c
2;
let c
5 be
RMembership_Func of c
2,c
3;
func c
4 (#) c
5 -> RMembership_Func of a
1,a
3 means :
Def3:
:: FUZZY_4:def 3
for b
1, b
2 being
set holds
(
[b1,b2] in [:a1,a3:] implies a
6 . [b1,b2] = sup (rng (min a4,a5,b1,b2)) );
existence
ex b1 being RMembership_Func of c1,c3 st
for b2, b3 being set holds
( [b2,b3] in [:c1,c3:] implies b1 . [b2,b3] = sup (rng (min c4,c5,b2,b3)) )
uniqueness
for b1, b2 being RMembership_Func of c1,c3 holds
( ( for b3, b4 being set holds
( [b3,b4] in [:c1,c3:] implies b1 . [b3,b4] = sup (rng (min c4,c5,b3,b4)) ) ) & ( for b3, b4 being set holds
( [b3,b4] in [:c1,c3:] implies b2 . [b3,b4] = sup (rng (min c4,c5,b3,b4)) ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines (#) FUZZY_4:def 3 :
for b
1, b
2, b
3 being non
empty set for b
4 being
RMembership_Func of b
1,b
2for b
5 being
RMembership_Func of b
2,b
3for b
6 being
RMembership_Func of b
1,b
3 holds
( b
6 = b
4 (#) b
5 iff for b
7, b
8 being
set holds
(
[b7,b8] in [:b1,b3:] implies b
6 . [b7,b8] = sup (rng (min b4,b5,b7,b8)) ) );
Lemma10:
for b1, b2, b3 being non empty set
for b4 being RMembership_Func of b1,b2
for b5, b6 being RMembership_Func of b2,b3
for b7, b8 being set holds
( b7 in b1 & b8 in b3 implies sup (rng (min b4,(max b5,b6),b7,b8)) = max (sup (rng (min b4,b5,b7,b8))),(sup (rng (min b4,b6,b7,b8))) )
theorem Th13: :: FUZZY_4:13
Lemma11:
for b1, b2, b3 being non empty set
for b4, b5 being RMembership_Func of b1,b2
for b6 being RMembership_Func of b2,b3
for b7, b8 being set holds
( b7 in b1 & b8 in b3 implies sup (rng (min (max b4,b5),b6,b7,b8)) = max (sup (rng (min b4,b6,b7,b8))),(sup (rng (min b5,b6,b7,b8))) )
theorem Th14: :: FUZZY_4:14
Lemma12:
for b1, b2, b3 being non empty set
for b4 being RMembership_Func of b1,b2
for b5, b6 being RMembership_Func of b2,b3
for b7, b8 being set holds
( b7 in b1 & b8 in b3 implies sup (rng (min b4,(min b5,b6),b7,b8)) <= min (sup (rng (min b4,b5,b7,b8))),(sup (rng (min b4,b6,b7,b8))) )
theorem Th15: :: FUZZY_4:15
Lemma13:
for b1, b2, b3 being non empty set
for b4, b5 being RMembership_Func of b1,b2
for b6 being RMembership_Func of b2,b3
for b7, b8 being set holds
( b7 in b1 & b8 in b3 implies sup (rng (min (min b4,b5),b6,b7,b8)) <= min (sup (rng (min b4,b6,b7,b8))),(sup (rng (min b5,b6,b7,b8))) )
theorem Th16: :: FUZZY_4:16
Lemma14:
for b1, b2, b3 being non empty set
for b4 being RMembership_Func of b1,b2
for b5 being RMembership_Func of b2,b3
for b6, b7 being set holds
( b6 in b1 & b7 in b3 implies sup (rng (min (converse b5),(converse b4),b7,b6)) = sup (rng (min b4,b5,b6,b7)) )
theorem Th17: :: FUZZY_4:17
theorem Th18: :: FUZZY_4:18
for b
1, b
2, b
3 being non
empty set for b
4, b
5 being
RMembership_Func of b
1,b
2for b
6, b
7 being
RMembership_Func of b
2,b
3for b
8, b
9 being
set holds
( b
8 in b
1 & b
9 in b
3 & ( for b
10 being
set holds
( b
10 in b
2 implies ( b
4 . [b8,b10] <= b
5 . [b8,b10] & b
6 . [b10,b9] <= b
7 . [b10,b9] ) ) ) implies
(b4 (#) b6) . [b8,b9] <= (b5 (#) b7) . [b8,b9] )
theorem Th19: :: FUZZY_4:19
definition
let c
1, c
2 be non
empty set ;
func Imf c
1,c
2 -> RMembership_Func of a
1,a
2 means :: FUZZY_4:def 4
for b
1, b
2 being
set holds
(
[b1,b2] in [:a1,a2:] implies ( ( b
1 = b
2 implies a
3 . [b1,b2] = 1 ) & ( not b
1 = b
2 implies a
3 . [b1,b2] = 0 ) ) );
existence
ex b1 being RMembership_Func of c1,c2 st
for b2, b3 being set holds
( [b2,b3] in [:c1,c2:] implies ( ( b2 = b3 implies b1 . [b2,b3] = 1 ) & ( not b2 = b3 implies b1 . [b2,b3] = 0 ) ) )
uniqueness
for b1, b2 being RMembership_Func of c1,c2 holds
( ( for b3, b4 being set holds
( [b3,b4] in [:c1,c2:] implies ( ( b3 = b4 implies b1 . [b3,b4] = 1 ) & ( not b3 = b4 implies b1 . [b3,b4] = 0 ) ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in [:c1,c2:] implies ( ( b3 = b4 implies b2 . [b3,b4] = 1 ) & ( not b3 = b4 implies b2 . [b3,b4] = 0 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines Imf FUZZY_4:def 4 :
theorem Th20: :: FUZZY_4:20
theorem Th21: :: FUZZY_4:21
Lemma17:
for b1, b2, b3 being non empty set
for b4 being RMembership_Func of b1,b2
for b5, b6 being set holds
( b5 in b3 & b6 in b2 implies sup (rng (min (Zmf b3,b1),b4,b5,b6)) = (Zmf b3,b2) . [b5,b6] )
theorem Th22: :: FUZZY_4:22
Lemma19:
for b1, b2, b3 being non empty set
for b4 being RMembership_Func of b1,b2
for b5, b6 being set holds
( b5 in b1 & b6 in b3 implies sup (rng (min b4,(Zmf b2,b3),b5,b6)) = (Zmf b1,b3) . [b5,b6] )
theorem Th23: :: FUZZY_4:23
theorem Th24: :: FUZZY_4:24