:: FUZZY_3 semantic presentation
definition
let c
1, c
2 be non
empty set ;
func Zmf c
1,c
2 -> RMembership_Func of a
1,a
2 equals :: FUZZY_3:def 1
chi {} ,
[:a1,a2:];
correctness
coherence
chi {} ,[:c1,c2:] is RMembership_Func of c1,c2;
by FUZZY_1:13;
func Umf c
1,c
2 -> RMembership_Func of a
1,a
2 equals :: FUZZY_3:def 2
chi [:a1,a2:],
[:a1,a2:];
correctness
coherence
chi [:c1,c2:],[:c1,c2:] is RMembership_Func of c1,c2;
by FUZZY_1:2;
end;
:: deftheorem Def1 defines Zmf FUZZY_3:def 1 :
:: deftheorem Def2 defines Umf FUZZY_3:def 2 :
theorem Th1: :: FUZZY_3:1
theorem Th2: :: FUZZY_3:2
theorem Th3: :: FUZZY_3:3
theorem Th4: :: FUZZY_3:4
for b
1, b
2 being non
empty set for b
3 being
RMembership_Func of b
1,b
2 holds
(
max b
3,
(Umf b1,b2) = Umf b
1,b
2 &
min b
3,
(Umf b1,b2) = b
3 &
max b
3,
(Zmf b1,b2) = b
3 &
min b
3,
(Zmf b1,b2) = Zmf b
1,b
2 )
theorem Th5: :: FUZZY_3:5
theorem Th6: :: FUZZY_3:6
theorem Th7: :: FUZZY_3:7