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151
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 Posted: Mon Oct 20, 2008 11:58 am Post subject: "Proving" fuzzy set theory as an extension of classic theory |
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Hi there,
I am having some trouble understanding an exercise that asks one to
"prove that Zadeh's proposal for Compliment extends the classic
definition of Compliment".
I would really love to understand fully how this can be done - I fear
I am completely missing the point on HOW one would go about proving
such a thing. I am from a programming background and consider myself
a novice at Maths...
I have queried my tutors for further info but while awaiting a
response I thought it would be worth asking the community here for a
"for dummies" step by step explanation which will hopefully open up
the whole concept of PROVING how fuzzy proposals extend classic set
operations.
Thanks in advance for any help on this that you might be able to
offer.
Gav |
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Dmitry A. Kazakov
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| Posted: Mon Oct 20, 2008 6:24 pm Post subject: Re: "Proving" fuzzy set theory as an extension of classic th |
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On Mon, 20 Oct 2008 04:58:05 -0700 (PDT), 151 wrote:
| Quote: | I have queried my tutors for further info but while awaiting a
response I thought it would be worth asking the community here for a
"for dummies" step by step explanation which will hopefully open up
the whole concept of PROVING how fuzzy proposals extend classic set
operations.
|
Step 1. Define a *injective* mapping from crisp values to fuzzy values.
Step 2. Take an operation and prove that for each combination of the
arguments, if you substituted crisp values of the arguments by their fuzzy
counterparts and applied the fuzzy equivalent operation, then the crisp
equivalent of the result would be the same value as if you applied the
crisp operation.
Step 3. Since the mapping was *injective* q.e.d.
Example:
1. F -> 0, T -> 1 (injection {F,T} -> [0,1])
2. /\ -> min
F /\ F = F, min (0, 0) = 0
F /\ T = F, min (0, 1) = 0
T /\ F = F, min (1, 0) = 0
T /\ T = T, min (1, 1) = 1
is a proof that min (Zadeh's "and") is an "extension" of /\.
For further details see category theory, which is basically all about such
stuff (relation between mathematical structures):
http://en.wikipedia.org/wiki/Category_theory
http://mathworld.wolfram.com/Category.html
--
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de |
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151
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| Posted: Tue Oct 21, 2008 11:11 am Post subject: Re: "Proving" fuzzy set theory as an extension of classic th |
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On 20 Oct, 14:24, "Dmitry A. Kazakov" <mail...@dmitry-kazakov.de>
wrote:
| Quote: | On Mon, 20 Oct 2008 04:58:05 -0700 (PDT), 151 wrote:
I have queried my tutors for further info but while awaiting a
response I thought it would be worth asking the community here for a
"for dummies" step by step explanation which will hopefully open up
the whole concept of PROVING how fuzzy proposals extend classic set
operations.
Step 1. Define a *injective* mapping from crisp values to fuzzy values.
Step 2. Take an operation and prove that for each combination of the
arguments, if you substituted crisp values of the arguments by their fuzzy
counterparts and applied the fuzzy equivalent operation, then the crisp
equivalent of the result would be the same value as if you applied the
crisp operation.
Step 3. Since the mapping was *injective* q.e.d.
Example:
1. F -> 0, T -> 1 (injection {F,T} -> [0,1])
2. /\ -> min
F /\ F = F, min (0, 0) = 0
F /\ T = F, min (0, 1) = 0
T /\ F = F, min (1, 0) = 0
T /\ T = T, min (1, 1) = 1
is a proof that min (Zadeh's "and") is an "extension" of /\.
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Dmitry! Thank you very much indeed for your reply. Very helpful
information (I wish my notes had been as succinct!)
If I can trouble you a bit further -I see now how this result matches
what one would see with classical operations (namely the same truth
values) assuming one uses the extremes of the possible values from 0
to 1.
however you state that:
"Since the mapping was *injective* q.e.d."
After reading your reply I looked into injective mappings but I fear I
still don't grasp WHY this *proves* the fact in question?
Sorry, I know these are probably quite naive questions...
thanks again,
Gav |
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Dmitry A. Kazakov
Guest
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| Posted: Tue Oct 21, 2008 7:41 pm Post subject: Re: "Proving" fuzzy set theory as an extension of classic th |
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On Tue, 21 Oct 2008 04:11:42 -0700 (PDT), 151 wrote:
| Quote: | After reading your reply I looked into injective mappings but I fear I
still don't grasp WHY this *proves* the fact in question?
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Because it is how you would define the word "extension." It would be like
"containing a subset of same algebraic structure."
Defining that more or less formally, you would come to an injective mapping
to identify the subset and a set of operations to describe the structure
(of Boolean algebra).
--
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de |
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