Crisp set

crisp set in soft computing

1. CRISP SET BY T.DEEPIKA M.SC(INFO TECH ) NADAR SARASWATHI COLLEGE OF ARTS ANDSCIENCE

2. INTRODUCTION Classical Set theory also termed as crisp set theory . It is also the fundamental to the study of fuzzy sets. Theory of Crisp set had its roots of boolean logic

3. Classical /boolean logic crisp set cont…

4. In crisp set we have only two options that is yes and no . For example When we ask question .Is water colourless? In crisp set we tell only yes or no.

5. Universe of discourse:- The Universe of Discourse is also known as the Universal Set There is reference to a particular context contains all possible elements having the same characteristic and from which sets can be performed. We denoted E as universal set

6. Example :- The universal set of all students in a university. The universal set of all numbers in euclidean space. E

7. Set :- A set is a well defined collection objects . well defines means the objects either belongs to or not belongs to in the set. Example: A={a1,a2,………a n } Where a1,a2……… are called the members of the set. A set is known as list form.

8. A Set also be defined based on the properties the numbers have to satisfy. In such case ,a set a is defined as A={X|P(x)} P(x)->stands for the property p. This satisfies the member x.

9. Venn diagram:- E Venn diagram are pictorial representation to denote a set. A

10. Membership:- An element x is said to be a member of a set A if x belongs to the set A. The membership is indicated by And is pronounced “belongs to”. Thus x A means x belongs to A and x A means x does not belong to A.

11. Example:- A ={4,5,6,7,8,10}, X=3 and y=4. Each element either belongs to or does not belong to a set. The concept of membership is definite and therefore crisp.

12. Cardinality:- The number of elements in a set is called its cardinality . Cardinality of a set A is denoted as n(A). Example: If A={4,5,6,7} then |A|=4.

13. Family of set:- A set whose member are sets themselves,is referred to as a family of set. Example:- A={{1,3,5},{2,4,6},{5,10}} is a set.

14. Null set/empty set:- A set is said to be a null set or empty set if it has no member. A null set is indicated as ф or{} and indicates an impossible event. Example:- The set of all prime minister who are below 15 year of age.

15. Singleton set:- A set with a single element is called a singleton set. A singleton set has cardinality of 1. Example:- if A={a},then |A|=1.

16. Subset:- Given sets A and B defined over E the universal set,A is said to be a subset of B if A is fully contained in B that is every element of A is in B. A B ->A is a subset of B. A is a proper subset of B. A is called the improper subset of B.

17. SUPERSET:- Given sets A and B on E the universal set,A is said to be a superset of B if every element of B is contained in A. A B->A is a superset of B. If A contains B and is equivalent to B.

18. Power set:- A power set of a set A is the set of all possible subsets that are derivable from A including null set. A power set is indicated as p(A) and has cardinality of |p(A)|=2 |4|.

19. Operation on crisp sets:- UNION(U):- The union of two sets A and B (AUB)is the set of all elements that belong to A or B or both. AUB={x/x A or x B} Example: A={a,b,c,1,2} and B={1,2,3,a,c} We get A U B={a,b,c,1,2,3}

20. Intersection( ):- The intersection of two sets A and B (A^B) is the set of all elements that belongs to A and B. A^B={x|x A and x B} Example: A={a,b,c,1,2} and B={1,2,3,a,c} We get A B={a,c,1,2}

21. Complement(c): The complement of a set A (A|A ) is the set of elements which are in E but not in A. A ={x/x A,x E} Example: X={1,2,3,4,5,6,7} and A={5,4,3} We get A ={1,2,6,7}

22. Difference(-):- The difference of the set A and B is A-B the set of all elements which are in A.but not in B. A-B={x|x A and x B} Example: A={ a,b,c,d,e } and B={ b,d } We get A-B={ a,c,e }

23. Properties of crisp set:- Commutativity->AUB=BUA A B=B A Associativity:->(AUB)UC=AU(B U C) (A B) C=A (B C) Distributivity:->A U(B C)=(A U B) (A U C) A (B U C)=(A B)U(A C) Idempotence :->A U A=A A A =A Law of absorption:-> A U (A B)=A,A (A U B)=A

24. Conclusion:

25. Thank you