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 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.
- FUZZY::=following,
Net
- Set::Sets= given.
- fuzzy_set(Set)::= Set>->[0..1],
[ Interval Notation in math_21_Order ]
- 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)).
- (above)|-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
[ MINMAX in math_21_Order ]
and more on lattices is in
- LATTICE::= See http://www.csci.csusb.edu/dick/maths/math_41_Two_Operators.html#Lattice.
(End of Net
FUZZY)
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. . . . . . . . . ( end of section Theories of Fuzzy Sets) <<Contents | End>>
[ 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 ]
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
- STANDARD::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html
Glossary
- 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).
- given::reason="I've been told that...", used to describe a problem.
- given::variable="I'll be given a value or object like this...", used to describe a problem.
- goal::theorem="The result I'm trying to prove right now".
- goal::variable="The value or object I'm trying to find or construct".
- 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.
- hyp::reason="I assumed this in my last Let/Case/Po/...".
- QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
- QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
- RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last
assumption (let) that you introduced.