.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(x<y , x , y),
	For a,b:Real, max(x) ::= if(x>y , 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