.Open Theories of Fuzzy Sets
An object is either in or out of a standard set. Membership is a sharp
concept with only two values: true and false. We can model standard sets
via a membership function maping objects into either 1(true)
or 0 (false). In Lofteh Zadeh's Fuzzy Set theory membership can be any value
from 0 to 1. A person can be only partly a member of the fuzzy set
of tall people. There has been a lot written about what
can be done by applying fuzzy logic.
.Hole
At first glance such thinking would lead to a probability theory. But
Zadeh created a new way of thinking by interpreting union and
intersection in a different way.
FUZZY::=following,
.Net
Set::Sets= given.
fuzzy_set(Set) ::= Set>->[0..1],
.See http://www.csci.csusb.edu/dick/maths/math_21_Order.html#Interval Notation
For A:fuzzy_set(Set), not A ::= x+>(1-A(x)).
For A,B:fuzzy_set(Set), A and B ::= x+>$min(A(x), B(y)).
For A,B:fuzzy_set(Set), A or B ::= x+>$max(A(x), B(y)).
()|-$LATTICE(fuzzy_set(Set), min, max) and Net{ complete}.
Here
For a,b:Real, min(x) ::= if(xy , x , y).
The properties of min and max in general are formulated in
.See http://www.csci.csusb.edu/dick/maths/math_21_Order.html#MINMAX
and more on lattices is in
LATTICE::=http://www.csci.csusb.edu/dick/maths/math_41_Two_Operators.html#Lattice.
.Close.Net FUZZY
.Hole
.Close Theories of Fuzzy Sets
. Also See
.See http://www/dick/maths/math_81_Probabillity.html
.See http://www/dick/maths/math_82_MultiSets_and_Bags.html
.See http://www/dick/maths/math_83_Fuzzy_Sets.html
.See http://www/dick/maths/math_84_Spectra.html