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. [click here if you can fill this 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.

1. FUZZY::=following,
   Net
   1. Set::Sets= given.
   2. fuzzy_set(Set)::= Set>->[0..1], [ Interval Notation in math_21_Order ]

   3. For A:fuzzy_set(Set), not A::= x+>(1-A(x)).
   4. For A,B:fuzzy_set(Set), A and B::= x+>min(A(x), B(y)).
   5. For A,B:fuzzy_set(Set), A or B::= x+>max(A(x), B(y)).
      
   6. (above)|-LATTICE(fuzzy_set(Set), min, max) and Net{ complete}.

      Here

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

      The properties of min and max in general are formulated in [ MINMAX in math_21_Order ] and more on lattices is in

   9. LATTICE::= See http://www.csci.csusb.edu/dick/maths/math_41_Two_Operators.html#Lattice.

   
   (End of Net FUZZY)
   
   

   [click here if you can fill this hole]

. . . . . . . . . ( end of section Theories of Fuzzy Sets) <<Contents | End>>

Also See

Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle = Net{radius:Positive Real, center:Point}.

For a complete listing of pages in this part of my site by topic see [ home.html ]

Notes on the Underlying Logic of MATHS

For a more rigorous description of the standard notations see