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CS344: Introduction to Artificial Intelligence

(associated lab: CS386)

Pushpak Bhattacharyya

CSE Dept., IIT Bombay

Lecture–1 to 6: Introduction; Fuzzy sets and logic; inverted pendulum

7th, 8th, 9th 14th, 15th , 21st Jan, 2013

(2)

Basic Facts

Faculty instructor: Dr. Pushpak Bhattacharyya (www.cse.iitb.ac.in/~pb)

TAship: Kashyap, Bibek, Samiulla, Lahari, Jayaprakash, Nikhil, Kritika and Shuvam

kan.pop@gmail.com, bibek.iitkgp@gmail.com, samiulla@cse.iitb.ac.in, lahari@cse.iitb.ac.in,

jayaprakash@cse.iitb.ac.in, nikhilkumar@cse.iitb.ac.in, kritika@cse.iitb.ac.in, shubhamg@cse.iitb.ac.in

Course home page

www.cse.iitb.ac.in/~cs344-2013 (will be up soon)

Venue: LCC 02, opp KR bldg

1 hour lectures 3 times a week: Mon-10.30, Tue-11.30, Thu- 8.30 (slot 2)

(3)

Perspective

(4)

AI Perspective (post-web)

Planning

Computer Vision

NLP

Expert Systems

Robotics

Search, Reasoning,

Learning IR

(5)

From Wikipedia

Artificial intelligence (AI) is the intelligence of machines and the branch of

computer science that aims to create it. Textbooks define the field as "the study and design of intelligent agents"[1] where an intelligent agent is a system that perceives its environment and takes actions that maximize its chances of

success.[2] John McCarthy, who coined the term in 1956,[3] defines it as "the science and engineering of making intelligent machines."[4]

The field was founded on the claim that a central property of humans, intelligence—

the sapience of Homo sapiens—can be so precisely described that it can be simulated by a machine.[5] This raises philosophical issues about the nature of the mind and limits of scientific hubris, issues which have been addressed by myth, fiction and philosophy since antiquity.[6] Artificial intelligence has been the subject of optimism,[7] but has also suffered setbacks[8] and, today, has become an essential part of the technology industry, providing the heavy lifting for many of the most difficult problems in computer science.[9]

AI research is highly technical and specialized, deeply divided into subfields that often fail to communicate with each other.[10] Subfields have grown up around particular institutions, the work of individual researchers, the solution of specific problems, longstanding differences of opinion about how AI should be done and the application of widely differing tools. The central problems of AI include such traits as reasoning, knowledge, planning, learning, communication, perception and the ability to move and manipulate objects.[11] General intelligence (or

"strong AI") is still a long-term goal of (some) research.[12]

(6)

Topics to be covered (1/2)

Search

General Graph Search, A*, Admissibility, Monotonicity

Iterative Deepening, α-β pruning, Application in game playing

Logic

Formal System, axioms, inference rules, completeness, soundness and consistency

Propositional Calculus, Predicate Calculus, Fuzzy Logic, Description Logic, Web Ontology Language

Knowledge Representation

Semantic Net, Frame, Script, Conceptual Dependency

Machine Learning

Decision Trees, Neural Networks, Support Vector Machines, Self Organization or Unsupervised Learning

(7)

Topics to be covered (2/2)

Evolutionary Computation

Genetic Algorithm, Swarm Intelligence

Probabilistic Methods

Hidden Markov Model, Maximum Entropy Markov Model, Conditional Random Field

IR and AI

Modeling User Intention, Ranking of Documents, Query Expansion, Personalization, User Click Study

Planning

Deterministic Planning, Stochastic Methods

Man and Machine

Natural Language Processing, Computer Vision, Expert Systems

Philosophical Issues

Is AI possible, Cognition, AI and Rationality, Computability and AI, Creativity

(8)

Foundational Points

Church Turing Hypothesis

Anything that is computable is computable by a Turing Machine

Conversely, the set of functions computed by a Turing Machine is the set of ALL and ONLY computable functions

(9)

Turing Machine

Finite State Head (CPU)

Infinite Tape (Memory)

(10)

Foundational Points (contd)

Physical Symbol System Hypothesis (Newel and Simon)

For Intelligence to emerge it is enough to manipulate symbols

(11)

Foundational Points (contd)

Society of Mind (Marvin Minsky)

Intelligence emerges from the interaction of very simple information processing units

Whole is larger than the sum of parts!

(12)

Foundational Points (contd)

Limits to computability

Halting problem: It is impossible to

construct a Universal Turing Machine that given any given pair <M, I> of Turing

Machine M and input I, will decide if M halts on I

What this has to do with intelligent computation? Think!

(13)

Foundational Points (contd)

Limits to Automation

Godel Theorem: A “sufficiently powerful”

formal system cannot be BOTH complete and consistent

“Sufficiently powerful”: at least as powerful as to be able to capture Peano’s Arithmetic

Sets limits to automation of reasoning

(14)

Foundational Points (contd)

Limits in terms of time and Space

NP-complete and NP-hard problems: Time for computation becomes extremely large as the length of input increases

PSPACE complete: Space requirement becomes extremely large

Sets limits in terms of resources

(15)

Two broad divisions of Theoretical CS

Theory A

Algorithms and Complexity

Theory B

Formal Systems and Logic

(16)

AI as the forcing function

Time sharing system in OS

Machine giving the illusion of attending simultaneously with several people

Compilers

Raising the level of the machine for better man machine interface

Arose from Natural Language Processing (NLP)

NLP in turn called the forcing function for AI

(17)

Allied Disciplines

Philosophy Knowledge Rep., Logic, Foundation of AI (is AI possible?)

Maths Search, Analysis of search algos, logic Economics Expert Systems, Decision Theory,

Principles of Rational Behavior

Psychology Behavioristic insights into AI programs Brain Science Learning, Neural Nets

Physics Learning, Information Theory & AI, Entropy, Robotics

Computer Sc. & Engg. Systems for AI

(18)

Goal of Teaching the course

Concept building: firm grip on foundations, clear ideas

Coverage: grasp of good amount of material, advances

Inspiration: get the spirit of AI,

motivation to take up further work

(19)

Resources

Main Text:

Artificial Intelligence: A Modern Approach by Russell & Norvik, Pearson, 2003.

Other Main References:

Principles of AI - Nilsson

AI - Rich & Knight

Knowledge Based Systems – Mark Stefik

Journals

AI, AI Magazine, IEEE Expert,

Area Specific Journals e.g, Computational Linguistics

Conferences

IJCAI, AAAI

Positively attend lectures!

(20)

Grading

Midsem

Endsem

Paper reading (possibly seminar)

Quizzes

(21)

Modeling Human Reasoning

Fuzzy Logic

(22)

Fuzzy Logic tries to capture the human ability of reasoning with imprecise information

Works with imprecise statements such as:

In a process control situation, “If the

temperature is moderate and the pressure is high, then turn the knob slightly right”

The rules have “Linguistic Variables”, typically adjectives qualified by adverbs (adverbs are hedges).

(23)

Alternatives to fuzzy logic model human reasoning (1/2)

Non-numerical

Non monotonic Logic

Negation by failure (“innocent unless proven guilty”)

Abduction (PQ AND Q gives P)

Modal Logic

New operators beyond AND, OR, IMPLIES, Quantification etc.

Naïve Physics

(24)

Abduction Example

If

there is rain (P)

Then

there will be no picnic (Q)

Abductive reasoning:

Observation: There was no picnic(Q)

Conclude : There was rain(P); in absence

of any other evidence

(25)

Alternatives to fuzzy logic model human reasoning (2/2)

Numerical

Fuzzy Logic

Probability Theory

Bayesian Decision Theory

Possibility Theory

Uncertainty Factor based on Dempster Shafer Evidence

Theory

(e.g. yellow_eyesjaundice; 0.3)

(26)

Linguistic Variables

Fuzzy sets are named by Linguistic Variables (typically adjectives).

Underlying the LV is a numerical quantity

E.g. For ‘tall’ (LV),

‘height’ is numerical quantity.

Profile of a LV is the

plot shown in the figure shown alongside.

µtall(h)

1 2 3 4 5 6 0

height h 1

0.4 4.5

(27)

Example Profiles

µrich(w)

wealth w

µpoor(w)

wealth w

(28)

Example Profiles

µA (x)

x

µA (x)

x Profile representing

moderate (e.g. moderately rich)

Profile representing extreme

(29)

Concept of Hedge

Hedge is an intensifier

Example:

LV = tall, LV1 = very tall, LV2 = somewhat tall

‘very’ operation:

µvery tall(x) = µ2tall(x)

‘somewhat’ operation:

µsomewhat tall(x) = √(µtall(x))

1

0 h

µtall(h)

somewhat tall tall

very tall

(30)

An Example

Controlling an inverted pendulum:

θ

d / dt

.

= angular velocity

Motor i=current

(31)

The goal: To keep the pendulum in vertical position (θ=0) in dynamic equilibrium. Whenever the pendulum departs from vertical, a torque is produced by sending a current ‘i’

Controlling factors for appropriate current Angle θ, Angular velocity θ.

Some intuitive rules

If θ is +ve small and θ. is –ve small then current is zero

If θ is +ve small and θ. is +ve small then current is –ve medium

(32)

-ve med -ve small Zero

+ve small +ve med

-ve med

-ve

small Zero +ve small

+ve med

+ve med

+ve small

-ve small

-ve med -ve

small +ve

small Zero

Zero

Zero

Region of interest

Control Matrix

θ. θ

(33)

Each cell is a rule of the form If θ is <> and θ. is <>

then i is <>

4 “Centre rules”

1. if θ = = Zero and θ. = = Zero then i = Zero

2. if θ is +ve small and θ. = = Zero then i is –ve small 3. if θ is –ve small and θ.= = Zero then i is +ve small 4. if θ = = Zero and θ. is +ve small then i is –ve small 5. if θ = = Zero and θ. is –ve small then i is +ve small

(34)

Linguistic variables 1. Zero

2. +ve small 3. -ve small

Profiles

ε2 2

3 ε3

+ve small -ve small

1

Quantity (θ, θ., i) zero

(35)

Inference procedure

1. Read actual numerical values of θ and θ.

2. Get the corresponding µ values µZero, µ(+ve small), µ(-ve small). This is called FUZZIFICATION

3. For different rules, get the fuzzy I-values from the R.H.S of the rules.

4. “Collate” by some method and get ONE current value. This is called DEFUZZIFICATION

5. Result is one numerical value of ‘i’.

(36)

if θ is Zero and dθ/dt is Zero then i is Zero

if θ is Zero and dθ/dt is +ve small then i is –ve small if θ is +ve small and dθ/dt is Zero then i is –ve small

if θ +ve small and dθ/dt is +ve small then i is -ve medium

ε2 2

3 ε3

+ve small -ve small

1

Quantity (θ, θ., i) zero

Rules Involved

(37)

Suppose θ is 1 radian and dθ/dt is 1 rad/sec µzero =1)=0.8 (say)

Μ+ve-small =1)=0.4 (say) µzero(dθ/dt =1)=0.3 (say)

µ+ve-small(dθ/dt =1)=0.7 (say)

ε2 2

3 ε3

+ve small -ve small

1

Quantity (θ, θ., i) zero

Fuzzification

1rad

1 rad/sec

(38)

Suppose θ is 1 radian and dθ/dt is 1 rad/sec µzero =1)=0.8 (say)

µ +ve-small =1)=0.4 (say) µzero(dθ/dt =1)=0.3 (say)

µ+ve-small(dθ/dt =1)=0.7 (say)

Fuzzification

if θ is Zero and dθ/dt is Zero then i is Zero min(0.8, 0.3)=0.3

hence µzero(i)=0.3

if θ is Zero and dθ/dt is +ve small then i is –ve small min(0.8, 0.7)=0.7

hence µ-ve-small(i)=0.7

if θ is +ve small and dθ/dt is Zero then i is –ve small min(0.4, 0.3)=0.3

hence µ-ve-small(i)=0.3

if θ +ve small and dθ/dt is +ve small then i is -ve medium min(0.4, 0.7)=0.4

hence µ-ve-medium(i)=0.4

(39)

2

3

-ve small

1

zero

Finding i

0.4

0.3 Possible candidates:

i=0.5 and -0.5 from the “zero” profile and µ=0.3

i=-0.1 and -2.5 from the “-ve-small” profile and µ=0.3 i=-1.7 and -4.1 from the “-ve-small” profile and µ=0.3 -4.1

-2.5

-ve small -ve medium

0.7

(40)

-ve small

zero

Defuzzification: Finding i by the centroid method

Possible candidates:

i is the x-coord of the centroid of the areas given by the blue trapezium, the green trapeziums and the black trapezium

-4.1

-2.5 -ve medium

Required i value Centroid of three trapezoids

(41)

Fuzzy Sets

(42)

Theory of Fuzzy Sets

Intimate connection between logic and set theory.

Given any set ‘S’ and an element ‘e’, there is a very natural predicate, µs(e) called as the belongingness predicate.

The predicate is such that,

µs(e) = 1, iff e ∈ S

= 0, otherwise

For example, S = {1, 2, 3, 4}, µs(1) = 1 and µs(5) = 0

A predicate P(x) also defines a set naturally.

S = {x | P(x) is true}

For example, even(x) defines S = {x | x is even}

(43)

Fuzzy Set Theory (contd.)

Fuzzy set theory starts by questioning the fundamental assumptions of set theory viz., the belongingness

predicate, µ, value is 0 or 1.

Instead in Fuzzy theory it is assumed that, µs(e) = [0, 1]

Fuzzy set theory is a generalization of classical set theory aka called Crisp Set Theory.

In real life, belongingness is a fuzzy concept.

Example: Let, T = “tallness”

µT (height=6.0ft ) = 1.0 µT (height=3.5ft) = 0.2

An individual with height 3.5ft is “tall” with a degree 0.2

(44)

Representation of Fuzzy sets

Let U = {x1,x2,…..,xn}

|U| = n

The various sets composed of elements from U are presented as points on and inside the n-dimensional hypercube. The crisp sets are the corners of the hypercube.

(1,0) (0,0)

(0,1) (1,1)

x1 x2

x1 x2

(x1,x2)

A(0.3,0.4)

µA(x1)=0.3 µA(x2)=0.4

Φ

U={x1,x2}

A fuzzy set A is represented by a point in the n-dimensional space as the point {µA(x1), µA(x2),……µA(xn)}

(45)

Degree of fuzziness

The centre of the hypercube is the most fuzzy set. Fuzziness decreases as one nears the

corners

Measure of fuzziness

Called the entropy of a fuzzy set

) ,

( /

) ,

( )

( S d S nearest d S farthest

E

Entropy

Fuzzy set Farthest corner

Nearest corner

(46)

(1,0) (0,0)

(0,1) (1,1)

x1 x2

d(A, nearest)

d(A, farthest) (0.5,0.5)

A

(47)

Definition

Distance between two fuzzy sets

| ) (

) (

| )

,

(

1 2

1 2

1 s i

n

i

i

s

x x

S S

d

L1 - norm

Let C = fuzzy set represented by the centre point d(c,nearest) = |0.5-1.0| + |0.5 – 0.0|

= 1

= d(C,farthest)

=> E(C) = 1

(48)

Definition

Cardinality of a fuzzy set

n

i

i

s x

s m

1

) ( )

( (generalization of cardinality of classical sets)

Union, Intersection, complementation, subset hood

) ( 1

)

(x s x

sc

U x

x x

x s s

s

s ( ) max( ( ), ( )),

2 1

2

1

U x

x x

x s s

s

s ( ) min( ( ), ( )),

2 1

2

1

(49)

Example of Operations on Fuzzy Set

Let us define the following:

Universe U={X1 ,X2 ,X3}

Fuzzy sets

A={0.2/X1 , 0.7/X2 , 0.6/X3} and

B={0.7/X1 ,0.3/X2 ,0.5/X3}

Then Cardinality of A and B are computed as follows:

Cardinality of A=|A|=0.2+0.7+0.6=1.5 Cardinality of B=|B|=0.7+0.3+0.5=1.5 While distance between A and B

d(A,B)=|0.2-0.7)+|0.7-0.3|+|0.6-0.5|=1.0

What does the cardinality of a fuzzy set mean? In crisp sets it means the number of elements in the set.

(50)

Example of Operations on Fuzzy Set (cntd.)

Universe U={X1 ,X2 ,X3}

Fuzzy sets A={0.2/X1 ,0.7/X2 ,0.6/X3} and B={0.7/X1 ,0.3/X2 ,0.5/X3}

A U B= {0.7/X1, 0.7/X2, 0.6/X3} A ∩ B= {0.2/X1, 0.3/X2, 0.5/X3} Ac = {0.8/X1, 0.3/X2, 0.4/X3}

(51)

Laws of Set Theory

The laws of Crisp set theory also holds for fuzzy set theory (verify them)

These laws are listed below:

Commutativity: A U B = B U A

Associativity: A U ( B U C )=( A U B ) U C

Distributivity: A U ( B ∩ C )=( A ∩ C ) U ( B ∩ C) A ∩ ( B U C)=( A U C) ∩( B U C)

De Morgan’s Law: (A U B) C= AC BC

(A ∩ B) C= AC U BC

(52)

Distributivity Property Proof

Let Universe U={x1,x2,…xn} piAU(B∩C)(xi)

=max[µA(xi), µ(B∩C)(xi)]

= max[µA(xi), min(µB(xi),µC(xi))]

qi(AUB) ∩(AUC)(xi)

=min[max(µA(xi), µB(xi)), max(µA(xi), µC(xi))]

(53)

Distributivity Property Proof

Case I: 0<µCBA<1

pi = max[µA(xi), min(µB(xi),µC(xi))]

= max[µA(xi), µC(xi)]=µA(xi)

qi =min[max(µA(xi), µB(xi)), max(µA(xi), µC(xi))]

= min[µA(xi), µA(xi)]=µA(xi)

Case II: 0<µCAB<1

pi = max[µA(xi), min(µB(xi),µC(xi))]

= max[µA(xi), µC(xi)]=µA(xi)

qi =min[max(µA(xi), µB(xi)), max(µA(xi), µC(xi))]

= min[µB(xi), µA(xi)]=µA(xi) Prove it for rest of the 4 cases.

(54)

Note on definition by extension and intension S1 = {xi|xi mod 2 = 0 } – Intension

S2 = {0,2,4,6,8,10,………..} – extension

(55)

How to define subset hood?

(56)

Meaning of fuzzy subset

Suppose, following classical set theory we say

if

Consider the n-hyperspace representation of A and B A

B

x x

x A

B( ) ( )

(1,1)

(1,0) (0,0)

(0,1)

x1 x2

A .B1

.B2 .B3

Region where B(x) A(x)

(57)

This effectively means CRISPLY

P(A) = Power set of A Eg: Suppose

A = {0,1,0,1,0,1,……….,0,1} – 104 elements B = {0,0,0,1,0,1,……….,0,1} – 104 elements

Isn’t with a degree? (only differs in the 2nd element) )

(A P B

A B

(58)

Subset operator is the “odd man” out

AUB, A∩B, Ac are all “Set Constructors” while A  B is a Boolean Expression or predicate.

According to classical logic

In Crisp Set theory A  B is defined as

x xA  xB

So, in fuzzy set theory A  B can be defined as

x µA(x) µB(x)

(59)

Zadeh’s definition of subsethood goes against the grain of fuzziness theory

Another way of defining A  B is as follows:

x µA(x) µB(x)

But, these two definitions imply that µP(B)(A)=1 where P(B) is the power set of B

Thus, these two definitions violate the fuzzy principle that every belongingness except Universe is fuzzy

(60)

Fuzzy definition of subset

Measured in terms of “fit violation”, i.e. violating the condition

Degree of subset hood S(A,B)= 1- degree of superset

=

m(B) = cardinality of B

=

) ( )

(x A x

B

) (

)) (

) ( ,

0 max(

1 m B

x x

x

A

B

x

B(x)

(61)

We can show that Exercise 1:

Show the relationship between entropy and subset hood Exercise 2:

Prove that

) ,

( )

(A S A Ac A Ac

E   

) ( /

) (

) ,

(B A m A B m B

S  

Subset hood of B in A

(62)

Fuzzy sets to fuzzy logic

Forms the foundation of fuzzy rule based system or fuzzy expert system Expert System

Rules are of the form If

then Ai

Where Cis are conditions

Eg: C1=Colour of the eye yellow C2= has fever

C3=high bilurubin A = hepatitis

Cn

C

C1 2 ....

(63)

In fuzzy logic we have fuzzy predicates Classical logic

P(x1,x2,x3…..xn) = 0/1 Fuzzy Logic

P(x1,x2,x3…..xn) = [0,1]

Fuzzy OR

Fuzzy AND

Fuzzy NOT

)) ( ), ( max(

) ( )

(x Q y P x Q y

P

)) ( ), ( min(

) ( )

(x Q y P x Q y

P

) ( 1

) (

~ P x P x

(64)

Fuzzy Implication

Many theories have been advanced and many expressions exist

The most used is Lukasiewitz formula

t(P) = truth value of a proposition/predicate. In fuzzy logic t(P) = [0,1]

t( ) = min[1,1 -t(P)+t(Q)]

P Q

Lukasiewitz definition of implication

(65)

Fuzzy Inferencing

Two methods of inferencing in classical logic

Modus Ponens

Given p and pq, infer q

Modus Tolens

Given ~q and pq, infer ~p

How is fuzzy inferencing done?

(66)

A look at reasoning

Deduction: p, p q|- q

Induction: p

1

, p

2

, p

3

, …|- for_all p

Abduction: q, p q|- p

Default reasoning: Non-monotonic reasoning: Negation by failure

If something cannot be proven, its negation is asserted to be true

E.g., in Prolog

(67)

Fuzzy Modus Ponens in terms of truth values

Given t(p)=1 and t(pq)=1, infer t(q)=1

In fuzzy logic,

given t(p)>=a, 0<=a<=1

and t(p>q)=c, 0<=c<=1

What is t(q)

How much of truth is transferred over the channel

p q

(68)

Lukasiewitz formula for Fuzzy Implication

t(P) = truth value of a proposition/predicate. In fuzzy logic t(P) = [0,1]

t( ) = min[1,1 -t(P)+t(Q)]

P Q Lukasiewitz definition of implication

(69)

Use Lukasiewitz definition

t(pq) = min[1,1 -t(p)+t(q)]

We have t(p->q)=c, i.e., min[1,1 -t(p)+t(q)]=c

Case 1:

c=1 gives 1 -t(p)+t(q)>=1, i.e., t(q)>=a

Otherwise, 1 -t(p)+t(q)=c, i.e., t(q)>=c+a-1

Combining, t(q)=max(0,a+c-1)

This is the amount of truth transferred over the channel pq

(70)

Two equations consistent

These two equations are consistent with each other

1 2

( , ) 1 ( , )

max(0, ( ) ( ))

1 where { , ,..., }

( )

( ( ) ( )) min(1,1 ( ( )) ( ( )))

i

i

B i A i

x U

n

B i

x U

B i A i B i A i

Sub B A Sup B A

x x

U x x x

x

t x x t x t x

 

 

(71)

Proof

Let us consider two crisp sets A and B

1 U

A B

2 U

B A

3 U

A B

4 U

A B

(72)

Proof (contd…)

Case I:

So,

( ) 1 only when ( ) 1 , ( ) ( ) 0

A xi B xi So B xi A xi



max(0, ( ) ( )) ( , ) 1

( )

1 0 1

( )

i

i

i

B i A i

x U

B i

x U

B i

x U

x x

Sub B A

x

x

 

 

(73)

Proof (contd…)

Thus, in case I these two equations are

consistent with each other (prove for other cases)

( ) ( ) 0

( ( ) ( )) min(1,1 ( ( ( )) ( ( )))) min(1,1 ( )) 1

B i A i

B i A i B i A i

Since x x

L t x x t x t x

ve



 

(74)

Proof of the fact that S(B,A) is consistent with crisp set theory

Case II (of the figure):

So,

( ) 1 if, ( ) 1 , ( ) ( ) 0

B xi A xi So B xi A xi



max(0, ( ) ( )) ( , ) 1

( )

i

i

B i A i

x U

B i

x U

x x

Sub B A

x

 

(75)

( ) ( ) 0

( , ) 1 1 ( ) , ( )

0 ( , ) 1

B i A i

A i

B i

x x

Sub B A x

x Sub B A

 

  

  

Proof (contd…)

Thus, in case II also, these two equations are consistent with each other.

(76)

Proof of case III

Case III:

So,

In This case, ( ) ( ) 0 for ( ) and ( ) ( ) 0 otherwise

B i A i i

B i A i

x x x B A

x x

  



max(0, ( ) ( )) ( , ) 1

( )

i

i

B i A i

x U

B i

x U

x x

Sub B A

x

 

(77)

Proof (contd…)

( ) ( ) 0 only for ( )

| | | |

( , ) 1

| | | |

B

x

i A

x

i

x

i

B A

B A A B

S B A

B B

  

 

  

Hence case III is also consistent across classical set theory and fuzzy set theory

(78)

Proof of case IV

Case IV:

So,

In This case, ( ) ( ) 1 for , ( ) ( ) 1 for

( ) ( ) 0 otherwise

B i A i i

B i A i i

B i A i

x x x A

x x x B

and x x

  

  

 

max(0, ( ) ( )) ( , ) 1

( )

i

i

B i A i

x U

B i

x U

x x

Sub B A

x

 

(79)

Proof (contd…)

( ) ( ) 1 only for <=0 otherwise,

| |

( , ) 1 0

| |

B

x

i A

x

i

x

i

B

and

S B A B

B

 

  

Thus, in case IV also, these two equations are consistent with each other.

Hence we can say that these two equations are consistent with each other in general.

References

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