Fuzzy Relations, Rules And Inferences
Fuzzy Relations, Rules and InferencesDebasis SamantaIIT Kharagpurdsamanta@iitkgp.ac.in06.02.2018Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.20181 / 64
Fuzzy RelationsDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.20182 / 64
Crisp relationsTo understand the fuzzy relations, it is better to discuss first crisprelation.Suppose, A and B are two (crisp) sets. Then Cartesian productdenoted as A B is a collection of order pairs, such thatA B {(a, b) a A and b B}Note :(1) A B 6 B A(2) A B A B (3)A B provides a mapping from a A to b B.The mapping so mentioned is called a relation.Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.20183 / 64
Crisp relationsExample 1:Consider the two crisp sets A and B as given below. A { 1, 2, 3, 4}B {3, 5, 7 }.Then, A B {(1, 3), (1, 5), (1, 7), (2, 3), (2, 5), (2, 7), (3, 3), (3, 5),(3, 7), (4, 3), (4, 5), (4, 7)}Let us define a relation R as R {(a, b) b a 1, (a, b) A B}Then, R {(2, 3), (4, 5)} in this case.We can represent the relation R in a matrix form as follows.1R 234Debasis Samanta (IIT Kharagpur)3570 1 000001 00 0 0 Soft Computing Applications06.02.20184 / 64
Operations on crisp relationsSuppose, R(x, y) and S(x, y ) are the two relations define over twocrisp sets x A and y BUnion:R(x, y) S(x, y ) max(R(x, y), S(x, y ));Intersection:R(x, y) S(x, y ) min(R(x, y), S(x, y ));Complement:R(x, y ) 1 R(x, y)Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.20185 / 64
Example: Operations on crisp relationsExample:Suppose, R(x, y) and S(x, y ) are the two relations define over twocrisp sets x A and y B 0 1 0 01 0 0 0 0 0 1 0 and S 0 1 0 0 ;R 0 0 0 1 0 0 1 0 0 0 0 00 0 0 1Find the following:1R S2R S3RDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.20186 / 64
Composition of two crisp relationsGiven R is a relation on X ,Y and S is another relation on Y ,Z .Then R S is called a composition of relation on X and Z which isdefined as follows.R S {(x, z) (x, y) R and (y , z) S and y Y }Max-Min CompositionGiven the two relation matrices R and S, the max-min composition isdefined as T R S ;T (x, z) max{min{R(x, y), S(y, z) and y Y }}Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.20187 / 64
Composition: CompositionExample:GivenX {1, 3, 5}; Y {1, 3, 5}; R {(x, y) y x 2}; S {(x, y) x y }Here, R and S is on X Y .Thus, we haveR {(1, 3), (3, 5)}S {(1, 3), (1, 5), (3, 5)}11R 35 351 0 1 0 0 0 1 and S 0 0 051 335Debasis Samanta (IIT Kharagpur)Soft Computing Applications5 0 1 1 0 0 1 0 0 011Using max-min composition R S 3 35 0 0 1 0 0 0 0 0 006.02.20188 / 64
Fuzzy relationsFuzzy relation is a fuzzy set defined on the Cartesian product ofcrisp set X1 , X2 , ., XnHere, n-tuples (x1 , x2 , ., xn ) may have varying degree ofmemberships within the relationship.The membership values indicate the strength of the relationbetween the tuples.Example:X { typhoid, viral, cold } and Y { running nose, high temp,shivering }The fuzzy relation R is defined astyphoid viral 90.90.40.80.70.6Debasis Samanta (IIT Kharagpur) Soft Computing Applications06.02.20189 / 64
Fuzzy Cartesian productSupposeA is a fuzzy set on the universe of discourse X with µA (x) x XB is a fuzzy set on the universe of discourse Y with µB (y) y YThen R A B X Y ; where R has its membership function givenby µR (x, y ) µA B (x, y) min{µA (x), µB (y )}Example :A {(a1 , 0.2), (a2 , 0.7), (a3 , 0.4)}and B {(b1 , 0.5), (b2 , 0.6)}b1a1R A B a2a3Debasis Samanta (IIT Kharagpur)b2 0.2 0.2 0.5 0.6 0.4 0.4Soft Computing Applications06.02.201810 / 64
Operations on Fuzzy relationsLet R and S be two fuzzy relations on A B.Union:µR S (a, b) max{µR (a, b), µS (a, b)}Intersection:µR S (a, b) min{µR (a, b), µS (a, b)}Complement:µR (a, b) 1 µR (a, b)CompositionT R SµR S maxy Y {min(µR (x, y ), µS (y , z))}Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201811 / 64
Operations on Fuzzy relations: ExamplesExample:X (x1 , x2 , x3 ); Y (y1 , y2 ); Z (z1 , z2 , z3 );y1x1R x2x3y2 0.5 0.1 0.2 0.9 0.8 0.6z1S y1y2 z2z30.6 0.4 0.70.5 0.8 0.9z1z2 z3 0.5 0.4 0.5x2 0.5 0.8 0.9 R S x30.6 0.6 0.7µR S (x1 , y1 ) max{min(x1 , y1 ), min(y1 , z1 ), min(x1 , y2 ), min(y2 , z1 )} max{min(0.5, 0.6), min(0.1, 0.5)} max{0.5, 0.1} 0.5 and so on.x1Debasis Samanta (IIT Kharagpur) Soft Computing Applications06.02.201812 / 64
Fuzzy relation : An exampleConsider the following two sets P and D, which represent a set ofpaddy plants and a set of plant diseases. More preciselyP {P1 , P2 , P3 , P4 } a set of four varieties of paddy plantsD {D1 , D2 , D3 , D4 } of the four various diseases affecting the plantsIn addition to these, also consider another set S {S1 , S2 , S3 , S4 } bethe common symptoms of the diseases.Let, R be a relation on P D, representing which plant is susceptibleto which diseases, then R can be stated asD1P1R P2P3P4 0.6 0.1 0.90.9Debasis Samanta (IIT Kharagpur)D2D30.60.20.30.80.90.90.40.4D4 0.80.8 0.8 0.2Soft Computing Applications06.02.201813 / 64
Fuzzy relation : An exampleAlso, consider T be the another relation on D S, which is given byS1D1S D2D3D4 0.1 1.0 0.00.9S2S30.21.00.01.00.70.40.50.8S4 0.90.6 0.9 0.2Obtain the association of plants with the different symptoms of thedisease using max-min composition.Hint: Find R T , and verify thatS1P1R S P2P3P4Debasis Samanta (IIT Kharagpur) 0.8 0.8 0.80.8S2S30.80.80.80.80.80.80.80.7S4 0.90.9 0.9 0.9Soft Computing Applications06.02.201814 / 64
Fuzzy relation : Another exampleLet, R x is relevant to yand S y is relevant to zbe two fuzzy relations defined on X Y and Y Z , respectively,where X {1, 2, 3} ,Y {α, β, γ, δ} and Z {a, b}.Assume that R and S can be expressed with the following relationmatrices :α1R 23S βγδγδ 0.1 0.3 0.5 0.7 0.4 0.2 0.8 0.9 and0.6 0.8 0.3 0.2aαβ 0.9 0.2 0.50.7Debasis Samanta (IIT Kharagpur)b 0.10.3 0.6 0.2Soft Computing Applications06.02.201815 / 64
Fuzzy relation : Another exampleNow, we want to find R S, which can be interpreted as a derivedfuzzy relation x is relevant to z.Suppose, we are only interested in the degree of relevance between2 X and a Z . Then, using max-min composition,µR S (2, a) max{(0.4 0.9), (0.2 0.2), (0.8 0.5), (0.9 0.7)} max{0.4, 0.2, 0.5, 0.7} 0.7Rs Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201816 / 64
2D Membership functions : Binary fuzzy relations ( x, y )(Binary) fuzzy relations are fuzzy sets A B which map each elementin A B to a membership grade between 0 and 1 (both inclusive).Note that a membership function of a binary fuzzy relation can bedepicted with a 3D plot.Important: Binary fuzzy relations are fuzzy sets with two dimensionalMFs and so on.Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201817 / 64
2D membership function : An exampleLet, X R y (the positive real line)and R X Y ”y is much greater than x”The membership function of µR (x, y ) is defined as (y x)if y x4µR (x, y ) 0if y xSuppose, X {3, 4, 5} and Y {3, 4, 5, 6, 7}, then33R 454567 0 0.25 0.5 0.75 1.0 000.25 0.5 0.75 0000.25 0.5Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201818 / 64
Problems to ponder:How you can derive the following?If x is A or y is B then z is C;Given that1R1 : If x is A then z is c [R1 A C]2R2 : If y is B then z is C [R2 B C]Hint:You have given two relations R1 and R2 .Then, the required can be derived using the union operation of R1and R2Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201819 / 64
Fuzzy PropositionsDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201820 / 64
Two-valued logic vs. Multi-valued logicThe basic assumption upon which crisp logic is based - that everyproposition is either TRUE or FALSE.The classical two-valued logic can be extended to multi-valuedlogic.As an example, three valued logic to denote true(1), false(0) andindeterminacy ( 21 ).Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201821 / 64
Two-valued logic vs. Multi-valued logicDifferent operations with three-valued logic can be extended as shownin the following truth table:a000121212111b0121012 00001212100121211 012112121111 a111 111 1121212121212110012121111112012Fuzzy connectives used in the above table are:AND ( ), OR ( ), NOT ( ), IMPLICATION ( ) and EQUAL ( ).Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201822 / 64
Three-valued logicFuzzy connectives defined for such a three-valued logic better can bestated as follows:Symbol ConnectiveNOTORANDIMPLICATION EQUALITYDebasis Samanta (IIT Kharagpur)Usage PP QP Q(P Q) or( P Q)(P Q) or[(P Q) (Q P)]Soft Computing ApplicationsDefinition1 T (P)max{T(P), T(Q) }min{ T(P),T(Q) }max{(1 - T(P)),T(Q) }1 T (P) T (Q) 06.02.201823 / 64
Fuzzy propositionExample 1:P : Ram is honest1T(P) 0.0: Absolutely false2T(P) 0.2: Partially false3T(P) 0.4: May be false or not false4T(P) 0.6: May be true or not true5T(P) 0.8: Partially true6T(P) 1.0: Absolutely true.Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201824 / 64
Example 2 :Fuzzy propositionP : Mary is efficient ; T(P) 0.8;Q : Ram is efficient ; T(Q) 0.61Mary is not efficient.T ( P) 1 T (P) 0.22Mary is efficient and so is Ram.T (P Q) min{T (P), T (Q)} 0.63Either Mary or Ram is efficientT (P Q) maxT (P), T (Q) 0.84If Mary is efficient then so is RamT (P Q) max{1 T (P), T (Q)} 0.6Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201825 / 64
Fuzzy proposition vs. Crisp propositionThe fundamental difference between crisp (classical) propositionand fuzzy propositions is in the range of their truth values.While each classical proposition is required to be either true orfalse, the truth or falsity of fuzzy proposition is a matter of degree.The degree of truth of each fuzzy proposition is expressed by avalue in the interval [0,1] both inclusive.Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201826 / 64
Canonical representation of Fuzzy propositionSuppose, X is a universe of discourse of five persons.Intelligent of x X is a fuzzy set as defined below.Intelligent: {(x1 , 0.3), (x2 , 0.4), (x3 , 0.1), (x4 , 0.6), (x5 , 0.9)}We define a fuzzy proposition as follows:P : x is intelligentThe canonical form of fuzzy proposition of this type, P isexpressed by the sentence P : v is F .Predicate in terms of fuzzy set.P : v is F ; where v is an element that takes values v from someuniversal set V and F is a fuzzy set on V that represents a fuzzypredicate.In other words, given, a particular element v , this element belongsto F with membership grade µF (v ).Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201827 / 64
F (v )Graphical interpretation of fuzzy propositionP: v is FT(P)T(P) µF(v) for a v ε VvVFor a given value v of variable V in proposition P, T(P) denotes thedegree of truth of proposition P.Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201828 / 64
Fuzzy ImplicationsDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201829 / 64
Fuzzy ruleA fuzzy implication (also known as fuzzy If-Then rule, fuzzy rule,or fuzzy conditional statement) assumes the form :If x is A then y is Bwhere, A and B are two linguistic variables defined by fuzzy sets Aand B on the universe of discourses X and Y , respectively.Often, x is A is called the antecedent or premise, while y is B iscalled the consequence or conclusion.Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201830 / 64
Fuzzy implication : Example 1If pressure is High then temperature is LowIf mango is Yellow then mango is Sweet else mango is SourIf road is Good then driving is Smooth else traffic is HighThe fuzzy implication is denoted as R : A BIn essence, it represents a binary fuzzy relation R on the(Cartesian) product of A BDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201831 / 64
Fuzzy implication : Example 2Suppose, P and T are two universes of discourses representingpressure and temperature, respectively as follows.P { 1,2,3,4} and T { 10, 15, 20, 25, 30, 35, 40, 45, 50 }Let the linguistic variable High temperature and Low pressure aregiven asTHIGH {(20, 0.2), (25, 0.4), (30, 0.6), (35, 0.6), (40, 0.7), (45, 0.8), (50, 0.8)}PLOW (1, 0.8), (2, 0.8), (3, 0.6), (4, 0.4)Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201832 / 64
Fuzzy implications : Example 2Then the fuzzy implication If temperature is High then pressureis Low can be defined asR : THIGH PLOW20 25 30where, R 80.80.20.40.60.60.60.60.60.20.40.40.40.40.40.4 Note : If temperature is 40 then what about low pressure?Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201833 / 64
Interpretation of fuzzy rulesIn general, there are two ways to interpret the fuzzy rule A B asA coupled with BA entails BDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201834 / 64
Interpretation as A coupled with BR :A B A B T-norm operator.RX YµA (x) µB (y) (x,y) ; where is called aT-norm operatorThe most frequently used T-norm operators are:Minimum : Tmin (a, b) min(a, b) a bAlgebric product : Tap (a, b) abBounded product : Tbp (a, b) 0 (a b 1) b 1 a ifa 1Drastic product : Tdp b if 0 if a, b 1Here, a µA (x) and b µB (y ). T is called the function of T-norm operator.Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201835 / 64
Interpretation as A coupled with BBased on the T-norm operator as defined above, we can automaticallydefine the fuzzy rule R : A B as a fuzzy set with two-dimentional MF:µR (x, y ) f (µA (x), µB (y)) f (a, b) with a µA (x) , b µB (y), and f isthe fuzzy implication function.Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201836 / 64
Interpretation as A coupled with BIn the following, few implications of R : A BMin operator:Rm A B RX YµA (x) µB (y ) (x,y) or fmin (a, b) a b[Mamdani rule]Algebric product operatorRRap A B X Y µA (x).µB (y) (x,y ) or fap (a, b) ab[Larsen rule]Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201837 / 64
Product OperatorsBounded product operatorRRRbp A B X Y µA (x) µB (y) (x,y ) X Y 0 (µA (x) µB (y ) 1) (x,y)or fbp 0 (a b 1)Drastic productR operatorRdp A B X Y µA (x)ˆ µB (y) (x,y) b 1 a ifa 1or fdp (a, b) b if 0 if otherwiseDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201838 / 64
Interpretation of A entails BThere are three main ways to interpret such implication:Material implication :R : A B Ā BPropositional calculus :R : A B Ā (A B)Extended propositional calculus :R : A B (Ā B̄) BDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201839 / 64
Interpretation of A entails BWith the above mentioned implications, there are a number of fuzzyimplication functions that are popularly followed in fuzzy rule-basedsystem.Zadeh’s arithmetic rule :RRza Ā B X Y 1 (1 µA (x) µB (y )) (x,y)orfza (a, b) 1 (1 a b)Zadeh’s max-min rule :RRmm Ā (A B) X Y (1 µA (x)) (µA (x) µB (y )) (x,y)orfmm (a, b) (1 a) (a b)Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201840 / 64
Interpretation of A entails BBoolean fuzzy ruleRRbf Ā B X Y (1 µA (x)) µB (x) (x,y)orfbf (a, b) (1 a) b;Goguen’s fuzzy rule:Rgf RX Y µA (x) µB (y ) (x,y ) where a b Debasis Samanta (IIT Kharagpur)Soft Computing Applications1 ifba ifa ba b06.02.201841 / 64
Example 3: Zadeh’s Max-Min ruleIf x is A then y is B with the implication of Zadeh’s max-min rule canbe written equivalently as :Rmm (A B) (Ā Y )Here, Y is the universe of discourse with membership values for ally Y is 1, that is , µY (y) 1 y Y .Suppose X {a, b, c, d} and Y {1, 2, 3, 4}and A {(a, 0.0), (b, 0.8), (c, 0.6), (d, 1.0)}B {(1, 0.2), (2, 1.0), (3, 0.8), (4, 0.0)} are two fuzzy sets.We are to determine Rmm (A B) (Ā Y )Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201842 / 64
Example 3: Zadeh’s min-max rule:The computation of Rmm (A B) (Ā Y ) is as follows:14da bc1Ā Y 3 000 0 0.2 0.8 0.8 0 0.2 0.6 0.6 0 and0.2 1.0 0.8 0aA B 2 bcdDebasis Samanta (IIT Kharagpur)234 1111 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0000Soft Computing Applications06.02.201843 / 64
Example 3: Zadeh’s min-max rule:Therefore,1aRmm (A B) (Ā Y ) bcdDebasis Samanta (IIT Kharagpur)234 1111 0.2 0.8 0.8 0.2 0.4 0.6 0.6 0.4 0.2 1.0 0.8 0Soft Computing Applications06.02.201844 / 64
Example 3 :X {a, b, c, d}Y {1, 2, 3, 4}Let, A {(a, 0.0), (b, 0.8), (c, 0.6), (d, 1.0)}B {(1, 0.2), (2, 1.0), (3, 0.8), (4, 0.0)}Determine the implication relation :If x is A then y is B1aHere, A B bcdDebasis Samanta (IIT Kharagpur)234 000 0 0.2 0.8 0.8 0 0.2 0.6 0.6 0 0.2 1.0 0.8 0Soft Computing Applications06.02.201845 / 64
Example 3 :1aand Ā Y bcd234 1111 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 00001aRmm (A B) (Ā Y ) bcd234 1111 0.2 0.8 0.8 0.2 0.4 0.6 0.6 0.4 0.2 1.0 0.8 0This R represents If x is A then y is BDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201846 / 64
Example 3 :IF x is A THEN y is B ELSE y is C.The relation R is equivalent toR (A B) (Ā C)The membership function of R is given byµR (x, y ) max[min{µA (x), µB (y )}, min{µĀ (x), µC (y )]Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201847 / 64
Example 4:X {a, b, c, d}Y {1, 2, 3, 4}A {(a, 0.0), (b, 0.8), (c, 0.6), (d, 1.0)}B {(1, 0.2), (2, 1.0), (3, 0.8), (4, 0.0)}C {(1, 0), (2, 0.4), (3, 1.0), (4, 0.8)}Determine the implication relation :If x is A then y is B else y is C1aHere, A B bcdDebasis Samanta (IIT Kharagpur)234 000 0 0.2 0.8 0.8 0 0.2 0.6 0.6 0 0.2 1.0 0.8 0Soft Computing Applications06.02.201848 / 64
Example 4:1aband Ā C cd1aR bcd 0 0.2 0.20.2Debasis Samanta (IIT Kharagpur)234 0 0.4 1.0 0.8 0 0.2 0.2 0.2 0 0.4 0.4 0.4 0 00020.40.80.61.034 1.0 0.80.8 0.2 0.6 0.4 0.8 0Soft Computing Applications06.02.201849 / 64
Interpretation of fuzzy implicationIf x is A then y is B{{If x is A then y is B else y is C{{Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201850 / 64
Fuzzy InferencesDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201851 / 64
Fuzzy inferencesLet’s start with propositional logic. We know the following inpropositional logic.1Modus Ponens : P, P Q, Q2Modus Tollens : P Q, Q , P3Chain rule : P Q, Q R , P RDebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201852 / 64
An example from propositional logicGiven1C D2 H (A B)3C D H4(A B) (R S)From the above can we infer R S?Similar concept is also followed in fuzzy logic to infer a fuzzy rule froma set of given fuzzy rules (also called fuzzy rule base).Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201853 / 64
Inferring procedures in Fuzzy logicTwo important inferring procedures are used in fuzzy systems :Generalized Modus Ponens (GMP)If x is A Then y is B0x is A————————————y is B0Generalized Modus Tollens (GMT)If x is A Then y is By is B0————————————0x is ADebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201854 / 64
Fuzzy inferring procedures00Here, A, B, A and B are fuzzy sets.00To compute the membership function A and B the max-min00composition of fuzzy sets B and A ,respectively with R(x, y)(which is the known implication relation) is to be used.Thus,0000B A R(x, y)A B R(x, y)Debasis Samanta (IIT Kharagpur)µB (y) max[min(µA0 (x), µR (x, y ))]µA (x) max[min(µB 0 (y), µR (x, y ))]Soft Computing Applications06.02.201855 / 64
Generalized Modus PonensGeneralized Modus Ponens (GMP)P : If x is A then y is BLet us consider two sets of variables x and y beX {x1 , x2 , x3 } and Y {y1 , y2 }, respectively.Also, let us consider the following.A {(x1 , 0.5), (x2 , 1), (x3 , 0.6)}B {(y1 , 1), (y2 , 0.4)}0Then, given a fact expressed by the proposition x is A ,0where A {(x1 , 0.6), (x2 , 0.9), (x3 , 0.7)}0derive a conclusion in the form y is B (using generalized modusponens (GMP)).Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201856 / 64
Example: Generalized Modus PonensIf x is A Then y is B0x is A————————————–0y is B00y1y2We are to find B A R(x, y ) where R(x, y) max{A B, A Y }x1A B x2x3 y1 0.5 0.4 1 0.4 and A Y 0.6 0.4x1x2x3y2 0.5 0.5 00 0.4 0.4Note: For A B, µA B (x, y ) min(µA x, µB (y))Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201857 / 64
Example: Generalized Modus Ponensyx1R(x, y ) (A B) (A y) x2x3y12 0.5 0.5 1 0.4 0.6 0.4 0Now, A {(x1 , 0.6), (x2 , 0.9), (x3 , 0.7)}00Therefore, B A R(x, y) 0.5 0.5 0.6 0.9 0.7 1 0.4 0.9 0.50.6 0.400Thus we derive that y is B where B {(y1 , 0.9), (y2 , 0.5)}Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201858 / 64
Example: Generalized Modus TollensGeneralized Modus Tollens (GMT)P:Q:If x is A Then y is By is B0——————————————0x is ADebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201859 / 64
Example: Generalized Modus TollensLet sets of variables x and y be X {x1 , x2 , x3 } and y {y1 , y2 },respectively.Assume that a proposition If x is A Then y is B given whereA {(x1 , 0.5), (x2 , 1.0), (x3 , 0.6)} and B {(y1 , 0.6), (y2 , 0.4)}Assume now that a fact expressed by a proposition y is B is given0where B {(y1 , 0.9), (y2 , 0.7)}.0From the above, we are to conclude that x is A . That is, we are to0determine ADebasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201860 / 64
Example: Generalized Modus TollensWe first calculate R(x, y) (A B) (A y )yx1R(x, y) x2x3y12 0.5 0.5 1 0.4 0.6 0.4 00Next, we calculate A B R(x, y )y10x1 A 0.9 0.7 x2x3y2 0.5 0.5 1 0.4 0.5 0.9 0.60.6 0.40Hence, we calculate that x is A where0A [(x1 , 0.5), (x2 , 0.9), (x3 , 0.6)]Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201861 / 64
PracticeApply the fuzzy GMP rule to deduce Rotation is quite slowGiven that :If temperature is High then rotation is Slow.temperature is Very HighLet,X {30, 40, 50, 60, 70, 80, 90, 100} be the set of temperatures.Y {10, 20, 30, 40, 50, 60} be the set of rotations per minute.Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201862 / 64
PracticeThe fuzzy set High(H), Very High (VH), Slow(S) and Quite Slow (QS)are given below.H {(70, 1), (80, 1), (90, 0.3)}VH {(90, 0.9), (100, 1)}S {(30, 0.8), (40, 1.0), (50, 0.6)}QS {(10, 1), (20, 0.8)}1If temperature is High then the rotation is Slow.R (H S) (H Y )2temperature is Very HighThus, to deduce ”rotation is Quite Slow”, we make use the compositionrule QS VH R(x, y )Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201863 / 64
Any questions?Debasis Samanta (IIT Kharagpur)Soft Computing Applications06.02.201864 / 64
2D Membership functions : Binary fuzzy relations (Binary) fuzzy relations are fuzzy sets A B which map each element in A B to a membership grade between 0 and 1 (both inclusive). Note that a membership function of a binary fuzzy relation can be depicted with a 3D plot. (, )xy P Important: Binary fuzzy relations are fuzzy sets with two dimensional
ing fuzzy sets, fuzzy logic, and fuzzy inference. Fuzzy rules play a key role in representing expert control/modeling knowledge and experience and in linking the input variables of fuzzy controllers/models to output variable (or variables). Two major types of fuzzy rules exist, namely, Mamdani fuzzy rules and Takagi-Sugeno (TS, for short) fuzzy .
fuzzy controller that uses an adaptive neuro-fuzzy inference system. Fuzzy Inference system (FIS) is a popular computing framework and is based on the concept of fuzzy set theories, fuzzy if and then rules, and fuzzy reasoning. 1.2 LITERATURE REVIEW: Implementation of fuzzy logic technology for the development of sophisticated
ii. Fuzzy rule base: in the rule base, the if-then rules are fuzzy rules. iii. Fuzzy inference engine: produces a map of the fuzzy set in the space entering the fuzzy set and in the space leaving the fuzzy set, according to the rules if-then. iv. Defuzzification: making something nonfuzzy [Xia et al., 2007] (Figure 5). PROPOSED METHOD
Different types of fuzzy sets [17] are defined in order to clear the vagueness of the existing problems. D.Dubois and H.Prade has defined fuzzy number as a fuzzy subset of real line [8]. In literature, many type of fuzzy numbers like triangular fuzzy number, trapezoidal fuzzy number, pentagonal fuzzy number,
methods based on the fuzzy relation. These can be found in (10,12-161. Fuzzy clustering, based on fuzzy relations, was first proposed by Tamura et al. [12]. They proposed an N-step procedure by using the composition of fuzzy relations beginning with a reflexive and symmetric fuzzy relation R in X. They showed that there is an n such that
of fuzzy numbers are triangular and trapezoidal. Fuzzy numbers have a better capability of handling vagueness than the classical fuzzy set. Making use of the concept of fuzzy numbers, Chen and Hwang [9] developed fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) based on trapezoidal fuzzy numbers.
Fuzzy Logic IJCAI2018 Tutorial 1. Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set 2. . A possible fuzzy set short 10. Example II : Fuzzy set 0 1 5ft 11ins 7 ft height . Fuzzy logic begins by borrowing notions from crisp logic, just as
Am I My Brother's Keeper? is a project by British artist Kate Daudy, who has transformed a large UNHCR tent; previously home to a Syrian refugee family in Jordan’s Za’atari camp into a participatory art installation focussing on the concepts of home and identity. During the year and a half she spent researching the project, Daudy visited refugee camps in Jordan. There and across Europe and .
