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Hints and Answers to Selected Problems of the Exercises

Chapter 1. Introduction

1.1

Hint : See the first and second paragraph of section 1.2 (Fuzzy Systems).1.2 Hint : See Table 1.2 (Soft Computing Techniques) and the subsequent text.

Chapter 2. Fuzzy Sets

2.1. Hint : Apply Venn diagram.

2.3 & 2.4. Hint : Use a graph paper. For various values ofx, see the value of (x), and then mapthis value to the new value using the transformation function.

2.7 A= { 0, 1, 2, 3 },B= { 2, 3, 5 }

R= { ( 0, 2 ), ( 0, 3 ), ( 0, 5 ), ( 1, 2 ), ( 2, 3 ), ( 2, 5 ), ( 3, 2 ) }

S= { ( 0, 2 ), ( 0, 3 ), ( 0, 5 ), ( 1, 3 ), ( 2, 5 ), ( 3, 5 ) }

T= { ( 2, 0 ), ( 3, 0 ), ( 3, 2 ), ( 3, 4 ), ( 5, 0 ), ( 5, 2 ) }

2.9 See Example 2.28 (-cut).

2.10 F= 0.6/a+ 0.2/b+ 0.3/c+ 0.9/d

F0.2 = { a, b, c, d} and 0.2F0..2= 0.2/a+ 0.2/b+0.2/c+ 0.2/d

F0.3 = { a, c, d} and 0.3F0.3= 0.3/a+ 0.3/c+ 0.3/d

F0.6 = { a, d} and 0.6F0.6= 0.6/a+ 0.6/d

F0.9 = { d} and 0.9F0.9= 0.9/d

2.11 See Example 2.32 (Fuzzy Cardinality).

2.12 & 2.13 See Examples 2.32 (Fuzzy Cardinality) and 2.33 (Fuzzy Extension principle).

Chapter 3. Fuzzy Logic

3.1 Hint : Realize AND and OR using , and .

3.2 Hint : Apply truth table method.

3.3 & 3.4 See Example 3.3 (Validity of an argument).

3.5 Hint : Recall that a collection of statements is said to be consistent if they can all be true

simultaneously. Construct the truth table and check.

3.10 See Examples 3.19 (Zadehs interpretation of fuzzy rule)

3.11 & 3.12 See Examples 3.21 (Fuzzy reasoning with the help of Generalized Modus Ponens)

Chapter 4. Fuzzy Inference Systems

4.1 See Sections 4.7.1 (Fuzzy air conditioner controller) and 4.7.2 (Fuzzy cruise controller).

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Chapter 5. Rough Sets

5.1 Proof follows from the definitions of equivalence relations and indiscernibility.

5.2 LetxB(U-X). Then [x]BUX[x]BX[x]BX= x B (X) xU-

B (X). Therefore,B(U-X) U-B (X). Similarly proveB(U-X) U-B (X).

5.5 The reduced table is

# CustomerName

Gender

(GD)

Amount

(A)

Payment

Mode (P)

1 Mili F High CC

2 Bill M Low Cash

3 Rita F High CC

4 Pam F High CC

5 Maya F Medium Cash6 Bob M Medium CC

7 Tony M Low Cash8 Gaga F High CC

9 Sam M Low Cash10 Abu M Low Cash

. Considering the indiscernible set of objects { 1, 3, 4, 8 } we derive the rule : IF Gender = F

AND (Age = High) THEN (Payment Mode = Credit Card). Obtain the other rules similarly.

5.7 See Example 5.17 (Data Clustering).

Chapter 6. Artificial Neural Networks : Basic Concepts

6.2 3 input, 1 output net with w1= w2= w3= 1, and activation function

1, ify_in2y_out=f(y_in) =

0, otherwise

6.3 Hint : Draw two straight lines separating the setsAandB.

6.4 Hint : What is the equation of a plane that separates the two classes?

6.6 See Example 6.6 (Realizing the logical AND function through Hebb learning).

6.7 See Example 6.7 (Learning the logical AND function by a perceptron).

6.9 See Example 6.9 (Competitive learning through winner-takes-all strategy).

Chapter 7. Elementary Pattern Classifiers

7.1 Hint : Take the training data in bipolar form and then apply Hebb learning rule.

7.3 See Example 7.4 (ADALINE training for the AND-NOT function).

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7.5 See Example 7.5 (MADALINE training for the XOR function).

Chapter 8. Pattern Associators

8.1 An n-input auto-associative net can store at most n- 1 patterns.

So,

1. Check whether number of patterns

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include production rules such as very etc.

11.6 See Problem 11.7 (Solving the Satisfiability problem using AND-OR graph).

11.7 Hint : The production rules will be expanded as AND arcs. When there is a choice of

production rules, i.e. there are several rules with the same string on the left hand side, use

OR arcs.

11.10 See Problem 11.10 (Applying constraint satisfaction to solve cryptarithmetic puzzle).

11.11

See Problem 11.9 (Applying constraint satisfaction to solve crossword puzzle).

11.12 See the text on the Map colouring problem in Subsection 11.4.8.

11.13 See Problem 11.2 (Monkey and Banana problem)

Chapter 12. Advanced Search Strategies

12.1 Hint: The chromosomes will be a binary string of length nk, where n is the number of

nodes and k= ceiling( log 2n ). Given such a chromosome, it is divided into nparts each

consisting of kbits. The first k-bit substring encodes 1, the 2nd

k-bit string encodes node 2

and so on. As the functionfis to be minimized, we may take 1/fas the fitness function. The

initial population is generated randomly.

12.2 Hint : Encode the solutions as in Ex. 12.1. Use fdirectly as the energy function. Try with

Tmax= 100, Tmin= 0.01, = 0.8. Tune the parameters if necessary.