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Content

- Crisp Sets
- Fuzzy Sets
- Set-Theoretic Operations
- Extension Principle
- Fuzzy Relations

Dr Basil Hamed

Fuzzy set theory provides a means for representing uncertainties.

Natural Language is vague and imprecise.

Fuzzy set theory uses Linguistic variables, rather than quantitative variables to represent imprecise concepts.

Dr Basil Hamed

Fuzzy Logic is suitable to

Very complex models

Judgmental

Reasoning

Perception

Decision making

Dr Basil Hamed

Dr Basil Hamed

Crisp set has a unique membership function

A(x) = 1 x A

0 x A

A(x) {0, 1}

Fuzzy Set can have an infinite number of membership functions

A [0,1]

Dr Basil Hamed

Dr Basil Hamed

CLASSICAL SETS

Define a universe of discourse, X, as a collection of objects all having the same characteristics. The individual elements in the universe X will be denoted as x. The features of the elements in X can be discrete, or continuous valued quantities on the real line. Examples of elements of various universes might be as follows:

- the clock speeds of computer CPUs;
- the operating currents of an electronic motor;
- the operating temperature of a heat pump;
- the integers 1 to 10.

Dr Basil Hamed

Union:

A B = {x | x A or x B}

Intersection:

A B = {x | x A and x B}

Complement:

A’ = {x | x A, x X}

X – Universal Set

Set Difference:

A | B = {x | x A and x B}

Set difference is also denoted by A - B

Dr Basil Hamed

Union of sets A and B (logical or).

Intersection of sets A and B.

Dr Basil Hamed

A B = B A

A B = B A

A (B C) = (A B) C

A (B C) = (A B) C

A (B C) = (A B) (A C)

A (B C) = (A B) (A C)

A A = A

A A = A

A X = X

A X = A

A = A

A =

Dr Basil Hamed

Mapping of Classical Sets to Functions

Mapping is an important concept in relating set-theoretic forms to function-theoretic representations of information. In its most general form it can be used to map elements or subsets in one universe of discourse to elements or sets in another universe.

Dr Basil Hamed

Dr Basil Hamed

- A fuzzy set, is a set containing elements that have varying degrees of membership in the set.
- Elements in a fuzzy set, because their membership need not be complete, can also be members of other fuzzy sets on the same universe.
- Elements of a fuzzy set are mapped to a universe of membership values using a function-theoretic form.

Dr Basil Hamed

- An object has a numeric “degree of membership”
- Normally, between 0 and 1 (inclusive)
- 0 membership means the object is not in the set
- 1 membership means the object is fully inside the set
- In between means the object is partially in the set

Dr Basil Hamed

If U is a collection of objects denoted generically by x, then a fuzzy setA in U is defined as a set of ordered pairs:

membership

function

U : universe of

discourse.

Dr Basil Hamed

Characteristic function X, indicating the belongingness of x to the set A

X(x) = 1 x A

0 x A

or called membership

Hence,

A B XA B(x)

= XA(x) XB(x)

= max(XA(x),XB(x))

Note:Some books use + for , but still it is not ordinary addition!

Dr Basil Hamed

A B XA B(x)

= XA(x) XB(x)

= min(XA(x),XB(x))

A’ XA’(x)

= 1 – XA(x)

A’’ = A

Dr Basil Hamed

A B(x) = A(x) B(x)

= max(A(x), B(x))

A B(x) = A(x) B(x)

= min(A(x), B(x))

A’(x) = 1 - A(x)

De Morgan’s Law also holds:

(A B)’ = A’ B’

(A B)’ = A’ B’

But, in general

A A’

A A’

Dr Basil Hamed

Union of fuzzy sets A and B∼

.

Intersection of fuzzy sets Aand B∼

.

Dr Basil Hamed

A A’ = X A A’ = Ø

Excluded middle axioms for crisp sets. (a) Crisp set A and its complement; (b) crisp A ∪ A= X (axiom of excluded middle); and (c) crisp A ∩ A = Ø (axiom of contradiction).

Dr Basil Hamed

A A’ A A’

Excluded middle axioms for fuzzy sets are not valid. (a) Fuzzy set A∼ and its complement; (b) fuzzy A∪ A∼ = X (axiom of excluded middle); and (c) fuzzy A ∩ A = Ø (axiom of contradiction).

Dr Basil Hamed

Set-Theoretic Operations

Dr Basil Hamed

Examples of Fuzzy Set Operations

- Fuzzy union (): the union of two fuzzy sets is the maximum (MAX) of each element from two sets.
- E.g.
- A = {1.0, 0.20, 0.75}
- B = {0.2, 0.45, 0.50}
- A B = {MAX(1.0, 0.2), MAX(0.20, 0.45), MAX(0.75, 0.50)} = {1.0, 0.45, 0.75}

Dr Basil Hamed

Examples of Fuzzy Set Operations

- Fuzzy intersection (): the intersection of two fuzzy sets is just the MIN of each element from the two sets.
- E.g.
- A B = {MIN(1.0, 0.2), MIN(0.20, 0.45), MIN(0.75, 0.50)} = {0.2, 0.20, 0.50}

Dr Basil Hamed

Examples of Fuzzy Set Operations

A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e}

B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}

Complement:

= {0/a, 0.7/b, 0.8/c 0.2/d, 1/e}

Union:

A B = {1/a, 0.9/b, 0.2/c, 0.8/d, 0.2/e}

Intersection:

A B = {0.6/a, 0.3/b, 0.1/c, 0.3/d, 0/e}

Dr Basil Hamed

A B = B A

A B = B A

A (B C) = (A B) C

A (B C) = (A B) C

A (B C) = (A B) (A C)

A (B C) = (A B) (A C)

A A = A A A = A

A X = X A X = A

A = A A =

If A B C, then A C

A’’ = A

Dr Basil Hamed

Note (x) [0,1]

not {0,1} like Crisp set

A = {A(x1) / x1 + A(x2) / x2 + …}

= { A(xi) / xi}

Note: ‘+’ add

‘/ ’ divide

Only for representing element and its membership.

Also some books use (x) for Crisp Sets too.

Dr Basil Hamed

# courses a student may take in a semester.

appropriate

# courses taken

1

0.5

0

2

4

6

8

x : # courses

Example (Discrete Universe)Dr Basil Hamed

# courses a student may take in a semester.

appropriate

# courses taken

Example (Discrete Universe)Alternative Representation:

Dr Basil Hamed

x : age

Example (Continuous Universe)U : the set of positive real numbers

about 50 years old

Alternative Representation:

Dr Basil Hamed

Alternative Notation

U: discrete universe

U: continuous universe

Note that and integral signs stand for the union of membership grades; “ / ” stands for a marker and does not imply division.

Dr Basil Hamed

Fuzzy Disjunction

- AB max(A, B)
- AB = C "Quality C is the disjunction of Quality A and B"

- (AB = C) (C = 0.75)

Dr Basil Hamed

Fuzzy Conjunction

- AB min(A, B)
- AB = C "Quality C is the conjunction of Quality A and B"

- (AB = C) (C = 0.375)

Dr Basil Hamed

Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

- Determine degrees of membership:

Dr Basil Hamed

Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

0.7

- Determine degrees of membership:
- A = 0.7

Dr Basil Hamed

Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

0.9

0.7

- Determine degrees of membership:
- A = 0.7 B = 0.9

Dr Basil Hamed

Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

0.9

0.7

- Determine degrees of membership:
- A = 0.7 B = 0.9

- Apply Fuzzy AND
- AB = min(A, B) = 0.7

Dr Basil Hamed

Generalized Union/Intersection

- Generalized Union
Or called triangular norm.

- Generalized Intersection

t-norm

t-conorm

Or called s-norm.

Dr Basil Hamed

T-norms and S-norms

- And/OR definitions are called T-norms (S-norms)
- Duals of one another
- A definition of one defines the other implicitly

- Many different ones have been proposed
- Min/Max, Product/Bounded-Sum, etc.
- Tons of theoretical literature
- We will not go into this.

Dr Basil Hamed

Hypothesis : Engineers are mathematicians. Logical thinkers do not believe in magic. Mathematicians are logical thinkers.

Conclusion :Engineers do not believe in magic.

Let us decompose this information into individual propositions

P: a person is an engineer

Q: a person is a mathematician

R: a person is a logical thinker

S: a person believes in magic

The statements can now be expressed as algebraic propositions as

((PQ)(RS)(QR))(PS)

Dr Basil Hamed

Example:

If X = {1,2,3}

Y = {a,b,c}

R = { (1 a),(1 c),(2 a),(2 b),(3 b),(3 c) }

a b c

1 1 0 1

R = 2 1 1 0

3 0 1 1

Using a diagram to represent the relation

Dr Basil Hamed

The Real-Life Relation

- x is close to y
- xand y are numbers

- x depends on y
- x and y are events

- x and y look alike
- x and y are persons or objects

- If x is large, then y is small
- x is an observed reading and y is a corresponding action

Dr Basil Hamed

Fuzzy Relations

- Triples showing connection between two sets:
(a,b,#): a is related to b with degree #

- Fuzzy relations are set themselves
- Fuzzy relations can be expressed as matrices

…

Dr Basil Hamed

Composition

Let R be a relation that relates, or maps, elements from universe X to universe Y, and let S be a relation that relates, or maps, elements from universe Y to universe Z.

A useful question we seek to answer is whether we can find a relation, T, that relates the same elements in universe X that R contains to the same elements in universe Z that S contains. It turns out that we can find such a relation using an operation known as composition.

Dr Basil Hamed

- If R is a fuzzy relation on the space X x Y
- S is a fuzzy relation on the space Y x Z
- Then, fuzzy composition is T = R S
- There are two common forms of the composition operation:
- Fuzzy max-min composition
- T(xz) = (R(xy) s(yz))
- 2. Fuzzy max-production composition
- T(xz) = (R(xy) s(yz))
- Note:R S S R multiplication

y Y

y Y

Dr Basil Hamed

Y

Z

Max-Min CompositionR: fuzzy relation defined on X and Y.

S: fuzzy relation defined on Y and Z.

R 。S: the composition of R and S.

A fuzzy relation defined on X an Z.

Dr Basil Hamed

Y

Z

.

Max-Product CompositionR: fuzzy relation defined on X and Y.

S: fuzzy relation defined on Y and Z.

R。S: the composition of R and S.

A fuzzy relation defined on X an Z.

Dr Basil Hamed

Example:

y1 y2 z1 z2 z3

R = x1 0.7 0.5 S = y1 0.9 0.6 0.2

x2 0.8 0.4 y2 0.1 0.7 0.5

z1 z2 z3

Using max-min, T = x1 0.7 0.6 0.5

x2 0.8 0.6 0.4

z1 z2 z3

Using max-product, T = x1 0.63 0.42 0.25

x2 0.72 0.48 0.20

Dr Basil Hamed

Example 3.8 (Page 59)

Suppose we are interested in understanding the speed control of the DC shunt motor under no-load condition, as shown.

Dr Basil Hamed

Example 3.8

Initially, the series resistance Rse in should be kept in the cut-in position for the following reasons:

1. The back electromagnetic force, given by Eb= kNφ, where k is a constant of proportionality, N is the motor speed, and φ is the flux (which is proportional to input voltage, V ), is equal to zero because the motor speed is equal to zero initially.

2. We have V = Eb+ Ia(Ra + Rse), therefore Ia= (V − Eb)/(Ra + Rse), where Ia is the armature current and Ra is the armature resistance. Since Eb is equal to zero initially, the armature current will be Ia= V/(Ra + Rse), which is going to be quite large initially and may destroy the armature.

Dr Basil Hamed

Example 3.8

Let Rsebe a fuzzy set representing a number of possible values for series resistance, say snvalues, given as

and let Iabe a fuzzy set having a number of possible values of the armature current, say m values, given as

The fuzzy sets Rseand Iacan be related through a fuzzy relation, say R, which would allow for the establishment of various degrees of relationship between pairs of resistance and current.

Dr Basil Hamed

Example 3.8

Let Nbe another fuzzy set having numerous values for the motor speed, say vvalues, given as

Now, we can determine another fuzzy relation, say S, to relate current to motor speed, that is, Iato N.

Using the operation of composition, we could then compute a relation, say T, to be used to relate series resistance to motor speed, that is, Rseto N.

Dr Basil Hamed

Example 3.8

The operations needed to develop these relations are as follows – two fuzzy Cartesian products and one composition:

Dr Basil Hamed

Example 3.8

Suppose the membership functions for both series resistance Rseand armature current Iaare given in terms of percentages of their respective rated values, that is,

Dr Basil Hamed

Example 3.8

The following relation then result from use of the Cartesian product to determine R:

Dr Basil Hamed

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