propositions are tautologies, contradictions, or neither.

a. P and (not Q)

b. (not P) or Q

c. (P and (P implies Q)) ! Q

d. ((P ! Q) and (Q ! R)) ! (P ! R)

Show that using the tautology (de Morgan™s theorem)

3.3

P or Q ¼ not((not P) and (not Q))

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55

3.7 QUESTIONS

we can obtain

a. (3.4) from (3.1)

b. (3.5) from (3.2)

c. (3.6) from (3.3)

Derive the formulas for k1 and k2 for a piecewise quadratic fuzzy number with

3.4

membership 0 at x ¼ a, 1 at x ¼ b, and 0 at x ¼ c with a , b , c. Evaluate k1

and k2 if a ¼ 0, b ¼ 1 and c ¼ 4, and test that m ¼ 0 at x ¼ (a þ b)/2 and

x ¼ (b þ c)/2. (See Section 3.3.2.)

Construct a truth table for the operator NAND: A NAND B ¼ NOT(A AND B)

3.5

using

a. Equation (3.1) for AND

b. Equation (3.3) for AND

Assume tv(P) and tv(Q) can be any value in [0, 1]. Show that

3.6

a. Equations (3.1) and (3.2) can produce different results.

b. Equations (3.3) and (3.6) can produce different results.

Evaluate equation (3.11) using (3.3) for AND and (3.6) for OR

3.7

Let A be a 3 ‚ 3 fuzzy matrix and we compute A2 ¼ A . A using min “max

3.8

composition as discussed in Section 3.4.1. Let A3 ¼ A2 . A, A4 ¼ A3 . A,

and so on.

a.

0 1

0 0:2 1

If A ¼ @ 0:4 0 1 A, then find A2 , A3 , A4 , A5 , . . .

0 1 0:3

b.

0 1

1 0 0:2

@ 0:4 0 A, then find A2 , A3 , . . .

If A ¼ 1

0 0:3 1

c. Make an educated guess on what happens to A2, A3, A4, . . . for all 3 ‚ 3

fuzzy matrices A.

Fuzzi¬cation in equation (3.31) was done using a fuzzy number for t. It can

3.9

also be done using crisp numbers for t, considering the crisp number as a sin-

gleton fuzzy number. Compute the discrete fuzzy set Temperature using

Figure 3.9 if

a. t ¼ 50

b. t ¼ 25

c. t ¼ 75

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56 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: I

Defuzzi¬cation, as described in Section 3.6.2, is somewhat complicated. To

3.10

speed up defuzzi¬cation assign to each fuzzy number its central value and

then ¬nd the centroid. In the example in 3.6.2, we obtain the discrete fuzzy set

0:42 0:88 0:14

F¼

0 50 100

from equation (3.31). Find the centroid of F and compare to the value of 41.8

obtained in Section 3.6.2.

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4 Fuzzy Logic, Fuzzy Sets, and

Fuzzy Numbers: II

4.1 INTRODUCTION

This chapter presents some topics in fuzzy systems theory more advanced than those

in Chapter 3. We begin with the algebra of fuzzy sets and fuzzy numbers, followed

by a discussion of fuzzy logical inference called approximate reasoning. A discus-

sion of the modifying words called hedges and a treatment of fuzzy propositions is

followed by a section on fuzzy arithmetic, which includes a section on the important

extension principle. The chapter concludes with a more complete treatment of fuzzy

comparisons than that given in Chapter 3. The problems at the end of the chapter are

designed to test your knowledge of basic fuzzy logic and fuzzy sets.

4.2 ALGEBRA OF FUZZY SETS

4.2.1 T-Norms and t-Conorms: Fuzzy AND and OR Operators

Given fuzzy sets A, B, C, . . . all fuzzy subsets of X, we wish to compute

A < B, B > C, and so on. What we use in fuzzy logic are the generalized AND

and OR operators from classical logic. They are called t-norms (for AND) and

t-conorms (for OR). We ¬rst de¬ne t-norms.

A t-norm T is a function from [0, 1] ‚ [0, 1] into [0, 1]. That is, if z ¼ T(x, y),

then x, y, and z all belong to the interval [0, 1]. All t-norms have the following

four properties:

T(x, 1) ¼ x (boundary)

1.

T(x, y) ¼ T( y, x) (commutativity)

2.

if y1 y2 , then T(x, y1 ) T(x, y2 ) (monotonicity)

3.

T(x, T( y, z)) ¼ T(T(x, y), z) (associativity)

4.

T-norms generalize the AND from classical logic. This means that tv(P AND

Q) ¼ T(tv(P), tv(Q)) for any t-norm and equations (4.1) “ (4.3) are all examples of

Fuzzy Expert Systems and Fuzzy Reasoning, By William Siler and James J. Buckley

ISBN 0-471-38859-9 Copyright # 2005 John Wiley & Sons, Inc.

57

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58 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: II

t-norms. The basic t-norms are

T m (x, y) ¼ min(x, y) (4:1)

T L (x, y) ¼ max(0, x þ y À 1) (4:2)

T p (x, y) ¼ xy (4:3)

and T Ã (x, y) de¬ned as x if y ¼ 1, y if x ¼ 1, 0 otherwise.

Tm is called the standard or Zadehian intersection, and is the one most commonly

employed; TL is the bounded difference intersection; Tp is the algebraic product; and

T Ã is the drastic intersection. It is well known that

TÃ TL Tp Tm (4:4)