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Tue Sep 18 15:18:48 PDT 2007


    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. [click here [socket symbol] 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,
      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}.


      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 [socket symbol] if you can fill this hole]

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

    Also See

    [ math_81_Probabillity.html ] [ math_82_MultiSets_and_Bags.html ] [ math_83_Fuzzy_Sets.html ] [ math_84_Spectra.html ]

    Notes on MATHS Notation

    Special characters are defined in [ intro_characters.html ] that also outlines the syntax of expressions and a document.

    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

    The notation used here is a formal language with syntax and a semantics described using traditional formal logic [ logic_0_Intro.html ] plus sets, functions, relations, and other mathematical extensions.

    For a more rigorous description of the standard notations see

  1. STANDARD::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html


  2. above::reason="I'm too lazy to work out which of the above statements I need here", often the last 3 or 4 statements. The previous and previous but one statments are shown as (-1) and (-2).
  3. given::reason="I've been told that...", used to describe a problem.
  4. given::variable="I'll be given a value or object like this...", used to describe a problem.
  5. goal::theorem="The result I'm trying to prove right now".
  6. goal::variable="The value or object I'm trying to find or construct".
  7. let::reason="For the sake of argument let...", introduces a temporary hypothesis that survives until the end of the surrounding "Let...Close.Let" block or Case.
  8. hyp::reason="I assumed this in my last Let/Case/Po/...".
  9. QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
  10. QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
  11. RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last assumption (let) that you introduced.