Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

1,577 views

Published on

fuzzy set theory with operations and fuzzy logic

No Downloads

Total views

1,577

On SlideShare

0

From Embeds

0

Number of Embeds

1

Shares

0

Downloads

120

Comments

0

Likes

3

No embeds

No notes for slide

- 1. What is Fuzzy? Fuzzy means not clear, distinct or precise; not crisp (well defined); blurred (with unclear outline).
- 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. Classical Set Vs Fuzzy set theory
- 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. 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. 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. 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. Fuzzy operations
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Example

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment