Module 3 - Lecture 06-07 - Fuzzy sets/logic

Question 1

Fuzzy sets and logic

What is an MF (membership function)?

Answer 1

A function that assigns a membership degree.

E.g. for any colour, how much does it belong to the set “red”?

Question 2

Fuzzy sets and logic

What is MF short for?

Answer 2

Membership function

Question 3

Fuzzy sets and logic

What is the difference between fuzzy sets and traditional sets?

Answer 3

Traditional sets are either 0 or 1, whereas fuzzy sets can be anything in-between. See image.

Question 4

Fuzzy sets and logic

How are fuzzy sets expressed? (Set notation)

Answer 4

As a set of ordered pairs of x values and membership values. (See image)

Question 5

Fuzzy sets and logic

What is mu in this image?

Answer 5

The membership function of A, which decided how much x belongs to A.

Question 6

Fuzzy sets and logic

What is the symbol for the membership function?

Answer 6

See image.

Question 7

Fuzzy sets and logic

Given the inputs in the image, what could the fuzzy set look like?

Answer 7

See image.

Question 8

Fuzzy sets and logic

Given the inputs in the image, what could the fuzzy set look like?

Answer 8

See image.

Question 9

Fuzzy sets and logic

What do plots of fuzzy sets look like? (Continuous and discrete)

Answer 9

See image

Question 10

Fuzzy sets and logic

What region is the “crossover points” of a membership function?

Answer 10

See image

Question 11

Fuzzy sets and logic

What region is the “alpha cuts” of a membership function?

Answer 11

See image

Question 12

Fuzzy sets and logic

What region is the “support” of a membership function?

Answer 12

See image

Question 13

Fuzzy sets and logic

What region is the “core” of a membership function?

Answer 13

See image

Question 14

Fuzzy sets and logic

How do you write the subset operator?

Answer 14

See image

Question 15

Fuzzy sets and logic

How do you write the complement operator?

Answer 15

See image

Question 16

Fuzzy sets and logic

How do you write the union operator?

Answer 16

@ See image

Question 17

Fuzzy sets and logic

@ How do you write the intersection operator?

Answer 17

See image

Question 18

Fuzzy sets and logic

Describe subsets visually.

Answer 18

See image

Question 19

Fuzzy sets and logic

Describe the NOT operation visually.

Answer 19

See image

Question 20

Fuzzy sets and logic

Describe the AND operation visually.

Answer 20

See image

Question 21

Fuzzy sets and logic

Describe the OR operation visually.

Answer 21

See image

Question 22

Fuzzy sets and logic

What is the formula for the triangular MF?

Answer 22

See image

Question 23

Fuzzy sets and logic

What is the formula for the trapezoidal MF?

Answer 23

See image

Question 24

Fuzzy sets and logic

What is the formula for the gaussian MF?

Answer 24

See image

Question 25

# Fuzzy sets and logic What is the formula for the generalized bell MF?

Answer 25

See image

Question 26

# Fuzzy sets and logic What is the formula for the sigmoidal MF?

Answer 26

See image

Question 27

# Fuzzy sets and logic How can you extend MFs?

Answer 27

E.g. by taking the product or absolute difference.

Question 28

# Fuzzy sets and logic What does the triangular MF look like?

Answer 28

See image

Question 29

# Fuzzy sets and logic What does the trapezoidal MF look like?

Answer 29

See image

Question 30

# Fuzzy sets and logic What does the gaussian MF look like?

Answer 30

See image

Question 31

# Fuzzy sets and logic What does the generalized bell MF look like?

Answer 31

See image

Question 32

# Fuzzy sets and logic What is an L-R MF?

Answer 32

See image. An MF that is piecewise-defined.

Question 33

# Fuzzy sets and logic What is an MF projection?

Answer 33

See image. Projection is when you have a multi-dimensional MF and project it down into a lower dimension.

Question 34

# Fuzzy sets and logic What is the formula for MF projection from a 2D MF down to X?

Answer 34

See image

Question 35

# Fuzzy sets and logic What are the two types of fuzzy complements?

Answer 35

Sugeno's and Yager's complement.

Question 36

# Fuzzy sets and logic What is the formula for Sugeno's complement?

Answer 36

See image

Question 37

# Fuzzy sets and logic What is the formula for Yager's complement?

Answer 37

See image

Question 38

# Fuzzy sets and logic What fuzzy complement is this formula? (See image)

Answer 38

Sugeno's

Question 39

# Fuzzy sets and logic What fuzzy complement is this formula?

Answer 39

Yager's

Question 40

# Fuzzy sets and logic What is the T-norm?

Answer 40

Fuzzy intersection

Question 41

# Fuzzy sets and logic What are some examples of T-norms? (MABD)

Answer 41

See image

Question 42

# Fuzzy sets and logic What are the monotonicity requirements for the T-norm?

Answer 42

See image

Question 43

# Fuzzy sets and logic What are the commutativity requirements for the T-norm?

Answer 43

See image

Question 44

# Fuzzy sets and logic What are the associativity requirements for the T-norm?

Answer 44

See image

Question 45

# Fuzzy sets and logic What are the boundary requirements for the T-norm?

Answer 45

See image

Question 46

# Fuzzy sets and logic What are the boundary requirements for the S-norm?

Answer 46

See image

Question 47

# Fuzzy sets and logic What are the monotonicity requirements for the S-norm?

Answer 47

See image

Question 48

# Fuzzy sets and logic What are the commutativity requirements for the S-norm?

Answer 48

See image

Question 49

# Fuzzy sets and logic What are the associativity requirements for the S-norm?

Answer 49

See image

Question 50

# Fuzzy sets and logic What law does the T- and S-norms satisfy?

Answer 50

DeMorgan's laws

Question 51

# Fuzzy sets and logic T-Norm DeMorgan's law: What is the equivalent of T(a, b)?

Answer 51

See image

Question 52

# Fuzzy sets and logic S-Norm DeMorgan's law: What is the equivalent of S(a, b)?

Answer 52

See image

Question 53

# Fuzzy sets and logic Describe the extension principle

Answer 53

If A is a fuzzy set on X, you might want to create B = f(A). then B = µ_B(x_i) / f(x_i) (Since A = {( x, µ_A(x)) | x \in X}, you have access to x, but afterwards B = {( f(x), µ_B(x)) | x \in X})

Question 54

# Fuzzy sets and logic What's the notation for a fuzzy relation?

Answer 54

See image

Question 55

# Fuzzy sets and logic What are some examples of fuzzy relations?

Answer 55

See image

Question 56

# Fuzzy sets and logic What is a linguistic variable?

Answer 56

A variable that can take linguistic values (words). E.g. Age is OLD.

Question 57

# Fuzzy sets and logic What's the name of a variable like this: Age is OLD?

Answer 57

That's a linguistic variable.

Question 58

# Fuzzy sets and logic What is a linguistic value?

Answer 58

It's a fuzzy set.

Question 59

# Fuzzy sets and logic Are linguistic values fuzzy sets?

Answer 59

Yes

Question 60

# Fuzzy sets and logic Are linguistic variables fuzzy sets?

Answer 60

No

Question 61

# Fuzzy sets and logic What is a term set?

Answer 61

A term set is a collection of linguistic values.

Question 62

# Fuzzy sets and logic What is this an example of? (See image)

Answer 62

A term set

Question 63

# Fuzzy sets and logic Describe the difference between primary and composite linguistic values.

Answer 63

See image

Question 64

# Fuzzy sets and logic What are the operations on linguistic variables? (CDI)

Answer 64

See image

Question 65

# Fuzzy sets and logic What's this an example of? If the road is slippery, then driving is dangerous.

Answer 65

A fuzzy rule.

Question 66

# Fuzzy sets and logic What are fuzzy rules?

Answer 66

A set of rules on the form: If X is A, then y is B. If the road is slippery, then driving is dangerous.

Question 67

# Fuzzy sets and logic What are the two ways of interpreting fuzzy rules? (CE)

Answer 67

See image

Question 68

# Fuzzy sets and logic What does it mean when A is coupled with B?

Answer 68

See image

Question 69

# Fuzzy sets and logic What does it mean when A entails B?

Answer 69

See image

Question 70

# Fuzzy sets and logic What is this an example of? (See image)

Answer 70

A is coupled with B

Question 71

# Fuzzy sets and logic What is this an example of? (See image)

Answer 71

A entails B

Question 72

# Fuzzy sets and logic What's the formula for fuzzy implication?

Answer 72

See image

Question 73

# Fuzzy sets and logic What is CRI short for?

Answer 73

Compositional rule of inference

Question 74

# Fuzzy sets and logic What is CRI (the compositional rule of inference)?

Answer 74

When you have x = a and some y = f(x), the CRI is how you find y = b.

Question 75

# Fuzzy sets and logic What is this an example of? (See image)

Answer 75

The compositional rule of inference.

Question 76

# Fuzzy sets and logic How do you write the complement of A? (As f(a))

Answer 76

N(a)

Question 77

# Fuzzy sets and logic How do you write Sugeno's complement of A? (As f(a))

Answer 77

N_s(a)

Question 78

# Fuzzy sets and logic How do you write Yager's complement of A? (as f(a))

Answer 78

N_w(a)

Question 79

# Fuzzy sets and logic What is the boundary requirement for the fuzzy complement?

Answer 79

N(0) = 1 and N(1) = 0

Question 80

# Fuzzy sets and logic What is the monotonicity requirement for the fuzzy complement?

Answer 80

N(a) > N(b) if a < b

Question 81

# Fuzzy sets and logic What is the involution requirement for the fuzzy complement?

Answer 81

N(N(a)) = a