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Fuzzy Foundations for EEs, CpEs, CSs. By P. D. Olivier, Ph.D., P.E. Fuzzy Foundations. Fuzzy “stuff” can be developed from two (albeit related) classical areas Classical (crisp) logic leads to Fuzzy Logic (path taken here)

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Fuzzy foundations for ees cpes css

Fuzzy Foundationsfor EEs, CpEs, CSs

By

P. D. Olivier, Ph.D., P.E.


Fuzzy foundations
Fuzzy Foundations

  • Fuzzy “stuff” can be developed from two (albeit related) classical areas

    • Classical (crisp) logic leads to Fuzzy Logic (path taken here)

    • Classical set theory leads to Fuzzy Set Theory (more common path, see book)

  • Logic and set theory are related since

    • AND and INTERSECTION are related

    • OR and UNION are related

    • NOT and COMPLEMENT are related


Classical logic

Invented by ancient Greeks, used by “classical scholars”, used by mathematicians

Every statement is either TRUEorFALSE

Statements can be combined with the logical connections AND and OR

A statement can be modified with NOT

Truth tables are used to evaluate the truth value (i.e. TRUEness or FALSEness) of a complicated statement

Logical IF – THEN statements are very important, used to express “THEOREMS”

Classical Logic


Truth table examples
Truth Table examples scholars”, used by mathematicians

Note: Logical IF premise THEN conclusion

not programming IF condition THEN action

{IF p THEN q}  {NOT(p)ORq}


Boolean algebra
Boolean Algebra scholars”, used by mathematicians

  • Based on Classical Logic

  • Truth values TRUE=T=1, FALSE=F=0

  • Mathematizes classical logic

    • Formulas evaluate truth values


Truth table example
Truth scholars”, used by mathematiciansTable example

  • TV(p) = truth value of p = 0 or 1

  • TV(pANDq) = min(TV(p),TV(q)) = TV(p)*TV(q) = …

  • TV(pORq) = max(TV(p),TV(q))

  • = TV(p)+TV(q)-TV(p)*TV(q) = …

  • TV(NOT(p))=1-TV(p)

  • Any logical expression can be expressed in terms of AND, OR, NOT.

TV{IF p THEN q} = TV{NOT(p)ORq}

= (1-TV(p))+TV(q)- (1-TV(p))*TV(q)


Fuzzy logic
Fuzzy Logic scholars”, used by mathematicians

  • Truth values are continuous between 0 and 1

  • Choose mathematical formulas for AND, OR, NOT

  • Compute truth values of complicated statements using chosen formulas

  • Are there other reasonable formulas of AND, OR, NOT?

  • What kind of vagueness does FL help with?

  • TV() function related to the characteristic function in classical set theory and classical logic

  • TV() function related to the membership function in FL


Types of vagueness
Types of Vagueness scholars”, used by mathematicians

  • Imprecision: Inaccurate measurement

  • Statistical: Precise, incomplete, measurements

  • Classification (membership in a set)

    • Determining membership in a group based on a measurement(s)

  • Fuzzy Logic/Set theory helps when set membership is not clear. Consider the set of TALL people. Determine if a given person is tall. Context, subjectivity.


  • Crisp set operations
    Crisp SET operations scholars”, used by mathematicians

    • An element x is either in a set or not in the set.

    • VENN diagrams

    • Union of A and B

    • Intersection of A and B

    • Complement of A


    Table 2 2
    Table 2.2 scholars”, used by mathematicians

    • Convert set equations to logical equations

      • CORRECTION: item one should read (A’)’=A


    Mathematizing crisp set theory
    Mathematizing CRISP set theory scholars”, used by mathematicians

    • Characteristic function

    • Complement

    • Intersection

    Others?

    • Union


    Fuzzy set theory
    Fuzzy Set theory scholars”, used by mathematicians

    • Characteristic functions become fuzzy membership functions

    • Fuzzy membership functions produce continuum of values between 0 and 1

    • Not just 0 or 1.

    • The value of the membership function at a point is the membership value of the point in the set.


    Fuzzy set theory logic
    Fuzzy scholars”, used by mathematiciansSet theory - Logic

    • Interpretation of Membership functions

      • truth value of a statement

      • Level of membership in a set

    • We will go back and forth between interpretations as convenient.

    • Fuzzy sets  Fuzzy membership functions


    Example 2 7 expensive cars
    Example 2.7: Expensive Cars scholars”, used by mathematicians

    • Logic statement:

      • Car X is an expensive car

    • Set theory statement

      • X is an element of the set of expensive cars

    • Consider Ferraris, Rolls Royce’s, Mercedes, BMWs, Buicks, Toyotas

      • Produce a membership function


    Example 2 8 natural numbers close to 6
    Example 2.8: Natural numbers close to 6 scholars”, used by mathematicians

    • Logic statement

      • n is a Natural number close to 6

    • Set theory statement

      • n is an element of the set of Natural numbers close to 6

    • Consider the natural numbers 3 … 9

    • Produce a membership function


    Typical fuzzy sets
    Typical Fuzzy sets scholars”, used by mathematicians

    • Increasing (s or gamma functions) pg 50

    • Decreasing (z or L functions)

    • Approximating (triangular/lambda, trapezoidal, bell)

    • Linguistic variables

      • Age

        • Old, young, middle aged, very old, very young

      • Temperature

        • Hot, cold, tepid, very hot, very cold, comfortable

      • Generic variable

        • NB, NM, NS, Z, PS, PM, PB


    Mathematical shorthand
    Mathematical shorthand scholars”, used by mathematicians

    • For all, or for every

    • There exists

    • Such that

    • With respect to

    : or s.t.

    w.r.t.


    2 1 5 properties of fuzzy sets see pp 52 54
    2.1.5 Properties of Fuzzy Sets (see pp 52-54) scholars”, used by mathematicians

    • Support

    • Width

      • sup

      • inf

    • Nucleus

    • Height

    • convexity

    Largest membership degree


    2 1 6 operations on fuzzy sets see pp 55 61
    2.1.6 Operations on Fuzzy Sets (see pp. 55-61) scholars”, used by mathematicians

    • Equality

    • Subset and strict subset

    • Superset and strict superset

    • Union, intersection and complement

    • Intersections are described by Triangular-norms (T-norms)

      • Archimedean

    • Unions are described by Triangular co-norms (S-norms)


    T norms generic intersection
    T-norms (generic intersection) scholars”, used by mathematicians

    • A triangular norm (T-norm) is a binary function (operator) that is

      • Commutative

      • Associative

      • Non dedreasing

      • T-norm identity is 1


    Archimedean t norms
    Archimedean T-norms scholars”, used by mathematicians

    • A T-norm that satisfies T-1 to T-4, together with


    S norms generic union
    S-norms (generic union) scholars”, used by mathematicians

    • A triangular co-norm (S-norm) is a binary function (operator) that is

      • Commutative

      • Associative

      • Non dedreasing

      • S-norm identity is 0


    Complement
    Complement scholars”, used by mathematicians

    • The complement function (operator) is a unary operator that has the following properties

    • Boundary

      values

    • Non-increasing

    • idempotent


    Exercises
    Exercises scholars”, used by mathematicians

    • Prove that min(a,b) and a*b are T-norms

    • Prove that max(a,b) and a+b-a*b are S-norms

    • Prove that min(a,b) and max(a,b) are conjugate T and S norms according to eq. 2.44

    • Prove that a*b and a+b-a*b are conjugate T and S norms according to eq. 2.44

    • Prove that 1-a is a complement operation

      PROVE means to demonstrate to a skeptic that the conclusion follows from the basic rules of mathematics.


    Classical to fuzzy relations
    Classical to Fuzzy Relations scholars”, used by mathematicians

    • A classical relation is a set of tuples

      • Binary relation (x,y)

      • Ternary relation (x,y,z)

      • N-ary relation (x1,…xn)

      • Connection with Cross product

      • Married couples

      • Nuclear family

      • Points on the circumference of a circle

      • Sides of a right triangle that are all integers


    Characteristic function
    Characteristic Function scholars”, used by mathematicians

    • Any set has a characteristic function.

    • A relation is a set of points

    • Review definition of characteristic function

    • Apply this definition to a set defined by a relation


    Properties of some binary relations
    Properties of some binary relations scholars”, used by mathematicians

    • Reflexive

    • Anti-reflexive

    • Symmetric

    • Anti-symmetric

    • Transitive

    • Equivalence

    • Partial order

    • Total order

    • Assignment: Classify: =,<,>,<=,>=


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