A general rule of thumb for understanding fuzzy math, carrying over from Boolean logic, is:

"and is inrsection is

In Classical Set Theory a

An

A fuzzy set **A** of

A *fuzzy relatiion* between two crisp sets ** X** and

Given fuzzy sets

A *fuzzy number* is a fuzzy subset of Â with bounded support and a convex continuous membership function that attains
1 for just one `x` Î Â (*ie.* a singleton core). A *fuzzy interval* is a fuzzy subset with a
a bounded support and a convex continuous membership function that attains unity over a core interval.
Convexity ensures that all a-cuts are single intervals, connectedTypical fuzzy number membership functions include triangular spikes and smooth bulges centered around a single
value `x`_{0} for which * A*(

This will generally be a fuzzy number for continous variables

If `a _{1}`,

A fuzzy number with a triangular spike membership function of the form

The product

(

Also only holding for postive

(

Consider a fuzzy rule of the form IF

IF

Viewed as a logical predicate and recalling from Classical Logic that (P Þ Q) = Q

Alternative implication predicate relations mentioned in the literature are

Mandini Implication:

Lukasiewicz Implication:

Product Implication:

Brouwerian Implication:

A compound antecedent such as

IF (`x` is __A___{1} AND `x` is __A___{2}) OR `x` is __A___{3} THEN can be replaced by
IF `x` is ((__A___{1} Ç __A___{2}) È __A___{3}) THEN where fuzzy set
(__A___{1} Ç __A___{2}) È __A___{3} has membership function

*((A_{1} Ç A_{2}) È A_{3})*(

More generally, we have

*Min* is the maximal t-norm in that *T*(`x`,`y`) £ *Min*{`x`,`y`}.
*Weak*(`x`,`y`) = *Min*{`x`,`y`} if *Max*{`x`,`y`}=1, 0 else is the minimal t-norm. Intermediary t-norms include the product `x``y` and
the Hamacher t-norm *H*_{g}(`x`,`y`) = (g + (1-g)(`x`+`y`-`x``y`))^{-1} `x``y` for g³| 0.

*Max* is the minimal t-conorm. *Strong*(`x`,`y`) = *Max*{`x`,`y`} if *Min*{`x`,`y`}=0, 1 else is the maximimal t-conorm.
Any given t-norm *T* has an associated derived t-conorm *S*(`x`,`y`) = 1 - *T*(1-`x`,1-`y`).
** Fuzzy Rules**

Consider *N* fuzzy rules:
IF `x`_{1} is __X___{1} THEN *z* is __Z___{1}
ALSO
IF `x`_{2} is __X___{2} THEN *z* is __Z___{2}
...
ALSO
IF *x _{N}* is

These are to be considered simultaneously true rules of fuzzy logic rather than sequential assignments. If

The wide variety of possible output aggregation and defuzzification strategies combined with the myriad choices of antecedent aggregation t-(co)norms combine to allow myriad alternate fuzzy logic schemeta. Combinned with the infinite potential for variouos antecedent fuzzy set membership functions we have an immensely rich modelling space.

For example, suppose we wish to model a function **¦**: Â × Â ® Â , perhaps given multiple (`x`,`y`,*z*=**¦**(`x`,`y`)) samples.
We can "cover" the domain of `x`,`y` with a number *R* of crisp real constant output fuzzy rules of the form

IF `x` is NEAR *x _{i}* AND

b

Using the product t-norm gives

b

Note that

There are two aproaches to fuzzing multivectors:

(i) Replacing some or all of the 2

(ii) Defining a

Easier perhaps to use Sugeno fuzzy rule schemata of the form IF x is

Consdider a fuzzy point having membership function m

If L =

For circle S, the a-cuts of "triangular" membership function m

Alternatively, we might use Gaussian membership function m

Suppose we wish to model a scalar field **¦**(** p**) . We could use axis aligned Gaussian ellipsoid patches as in the above
Â×Â®Â example but degree functions of the form

(b

where the b

Such approximations are chacterised by sets of scaled blades, which can be transformed to transform the field.

Next : Multivector Shapes

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Copyright (c) Ian C G Bell 2015, 2014

Web Source: www.iancgbell.clara.net/maths

Latest Edit: 07 Aug 2015.