Fuzzy logic

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fuzzy set theory with operations and fuzzy logic

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Fuzzy logic

  1. 1. What is Fuzzy?  Fuzzy means  not clear, distinct or precise;  not crisp (well defined);  blurred (with unclear outline).
  2. 2. Sets Theory  Classical Set: An element either belongs or does not belong to a sets that have been defined.  Fuzzy Set: An element belongs partially or gradually to the sets that have been defined.
  3. 3. Classical Set Vs Fuzzy set theory
  4. 4. Classical Set theory  Classical set theory represents all items elements, A={ a1,a2,a3,…..an} if elements ai (i=1,2,3,…n) of a set A are subset of universal set X, then set A can be represent for all elements x Є X by its characteristics function, 1 μA(x) = {0 if x Є X otherwise thus in classical set theory μA(x) has only values 0 (false) and 1( true). Such set are called crisp sets
  5. 5. Fuzzy Set Theory  Fuzzy set theory is an extension of classical set theory where element have varying degrees of membership. A logic based on the two truth values, True and false, is sometimes inadequate when describing human reasoning. Fuzzy logic uses the whole interval between 0 and 1 to describe human reasoning.  A fuzzy set is any set that allows its members to have different degree of membership, called membership function, in the interval [0,1].
  6. 6. Definition  A fuzzy set A, defines in the universal space X, is a function defined in X which assumes values in range [0,1].  A fuzzy set A is written as s set of pairs { x, A(x)} as A= {{x, A(x)}}, x in the set X. where x is element of universal space or set X and A(x) is the value of function A for this element.  Example: Set SMALL in set X consisting natural numbers <= 5.  Assume: SMALL(1)=1, SMALL(2)=1, SMALL(3)=0.9, SMALL(4)=0.6, SMALL(5)=0.4  Then set SMALL={ {1,1,},{2, 1},{3,0. 9},{4,0.6}, {5,0.4}}
  7. 7. Fuzzy V/s Crisp set Yes  Is water colourless? Crisp No Extremely honest(1) Very honest(0.85) Is Ram honest? Fuzzy Honest at time (0.4) Extremely dishonest(0)
  8. 8. Fuzzy operations
  9. 9. Union  The union of two fuzzy sets A and B is a new fuzzy set A U B also on X with membership function defined as μ A U B (x)= max (μ A (x) ,μ B (x))  Example: Let A be the fuzzy set of young people and B be the fuzzy set of middle-aged people. In discrete form, A=P(x1,0.5), (x2,0.7), (x3,0)} and B={(x1,0.8),( x2,0.2),( x3,1)} So find out A U B .
  10. 10. Use formula μ A U B (x1)= max (μ A (x1) ,μ B (x1)) = max(0.5,0.8) =0.8 μ A U B (x2)= max (μ A (x2) ,μ B (x2)) = max(0.7,0.2) =0.7 μ A U B (x3)= max (μ A (x3) ,μ B (x3)) = max(0,1) =1 So, A U B= {(x1,0.8, x2,0.7, x3,1)}
  11. 11. Intersection U  The union of two fuzzy sets A and B is a new fuzzy set A B also on X with membership function defined as μ A B (x)= min (μ A (x) ,μ B (x))  Example: Let A be the fuzzy set of young people and B be the fuzzy set of middle-aged people. In discrete form, A=P(x1,0.5), (x2,0.7), (x3,0)} and B={(x1,0.8),( x2,0.2),( x3,1)} So find out A B . U U
  12. 12. Use formula μ A B (x1)= min (μ A (x1) ,μ B (x1)) = max(0.5,0.8) =0.5 U B (x2)= U μA U μA min (μ A (x2) ,μ B (x2)) = max(0.7,0.2) =0.2 B (x3)= min (μ A (x3) ,μ B (x3)) = max(0,1) =0 So, A B= {(x1,0.5, x2,0.2, x3,0)} U
  13. 13. Complement  The complement of a fuzzy set A with membership function defined as μ A (x)= 1-μ A (x)  Example: Let A be the fuzzy set of young people complement “not young” is defined as Ac. In discrete form, for x1, x2, x3 A=P(x1,0.5), (x2,0.7), (x3,0)} So find out Ac.
  14. 14. Use formula μ A (x1)= 1- μ A (x1 ) = 1-0.5 =0.5 μ A (x1)= 1- μ A (x1 ) = 1-0.7 =0.3 μ A (x1)= 1- μ A (x1 ) = 1-0 =1 So, Ac= {(x1,0.5, x2,0.3, x3,1)}
  15. 15. Product of two fuzzy set  The product of two fuzzy sets A and B is a new fuzzy set A .B also on X with membership function defined as μ A.B (x)= μ A (x) μ B (x)  Example: Let A be the fuzzy set of young people and B be the fuzzy set of middle-aged people. In discrete form, A=P(x1,0.5), (x2,0.7), (x3,0)} and B={(x1,0.8),( x2,0.2),( x3,1)} So find out A.B
  16. 16. Use formula μ A .B (x1)= μ A (x1).μ B (x1) = 0.5 . 0.8 =0.040 μ A .B (x2)= μ A (x2).μ B (x2) = 0.7 . 0.2 =0.014 μ A .B (x3)= μ A (x3).μ B (x3) =0 . 1 =0 So, A .B= {(x1,0.040, x2,0.014, x3,0)}
  17. 17. Equality  The two fuzzy sets A and B is said to be equal(A=B) if μ A (x) =μ B (x)  Example: A=(x1,0.2), (x2,0.8)} B={(x1,0.6),( x2,0.8)} C={(x1,0.2),( x2,0.8)} μ A (x1) ≠μ B (x1) & μ A (x2) =μ B (x2) μ A (0.2) ≠μ B (0.6) & μ A (0.8) =μ B (0.8) so, A≠B μ A (x1) =μ c (x1) & μ A (x2) =μ c (x2) μ A (0.2) =μ c (0.2) & μ A (0.8) =μ c (0.8) so, A=C
  18. 18. Fuzzy Logic  Flexible machine learning technique  Mimicking the logic of human thought  Logic may have two values and represents two possible solutions  Fuzzy logic is a multi valued logic and allows intermediate values to be defined  Provides an inference mechanism which can interpret and execute commands  Fuzzy systems are suitable for uncertain or approximate reasoning
  19. 19. Fuzzy Logic  A way to represent variation or imprecision in logic  A way to make use of natural language in logic  Approximate reasoning  Definition of Fuzzy Logic: A form of knowledge representation suitable for notions that cannot be defined precisely, but which depend upon their contexts.  Superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth the truth values between "completely true & completely false".
  20. 20. Fuzzy Propositions A fuzzy proposition is a statement that drives a fuzzy truth value.  Fuzzy Connectives: Fuzzy connectives are used to join simple fuzzy propositions to make compound propositions. Examples of fuzzy connectives are:  Negation(-)  Disjunction(v)  Conjunction(^)  Impication( )
  21. 21. Example

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