Crisp sets

CRISP SETS OVERVIEW

1. CRISP SETS BY T.Deepika M.SC(COMPUTER SCIENCE) NADAR SARASWATHI COLLEGE OF ARTS AND SCIENCE

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

3. Cont.. Boolean logic Cr Crisp set

4. By using boolean logic,crisp set have only two options .i.e (YES or NO). For Example: 1.Is dog barks? ->> Yes. The dog barks. Here crisp set say only yes or no type answers.

5. Universe Of Discourse: Universe of discourse is also known as the Universal Set. It contains all elements having same characteristics. Universal Set is denoted by the symbol “ E”.

6. Set: A set is “ Well defined collection of objects ”. Example: A={X1,X2,X3,……………….X n } Where X1 ,X2 and X3 are called the members of the set. It is also known as “ LIST FORM”. A set is also be defined based on the properties of the numbers .

7. Venn Diagram: Venn diagram is a pictorial representation to denote a set. E A

8. 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 ε . X ε A means x belongs to A and x to A means x does not belong to A.

9. Example: A={1,2,3,4,5,6,7,8} X=9;Y=6. Each element from the set either belongs to or does not belongs to a set.And,therefore membership is definite.

10. Family Of Set: A set whose members are sets themselves , is referred to as a family of set. Example: A={{3,4,5},{1,2,3},{ 9,4}}

11. Subset: In a given set A and B defined over E the universal set,A is said to be a subset of B.(i.e)Every element of A is in B. A Contains B.Here, A is a subset of B. A is a proper subset of B. A is called the improper subset of B.

12. 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 Ͻ B. A is a superset of B. If A contains B and is equivalent to B.

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

14. Operations 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 Ϲ A or X Ϲ B}. Example: A={1,2,3,4,5} and B={a,b,c,d} AUB={a,b,c,d,1,2,3,4,5}

15. Intersection( ᴖ ): The intersection of two sets A and B (A ᴖ B) is the set of all elements that belongs to A and B. } AᴖB={x|x ε A and x ε B}\

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

17. 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 belongs to A and X belongs to B}. Example: A={a,b,c,d,e} and B={b,d} A-B={a,c,e}.

18. Properties of crisp set: Law of Commutativity: (A ∪ B) = (B ∪ A) (A ∩ B) = (B ∩ A) Law of Associativity: (A ∪ B) ∪ C = A ∪ (B ∪ C) (A ∩ B) ∩ C = A ∩ (B ∩ C) Law of Distributivity: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Idempotent Law: A ∪ Φ = A => A ∪ E = E A ∩ Φ = Φ => A ∩ E = A

19. Law of Absorption A ∪ (A ∩ B) = A A ∩ (A ∪ B) = A Law of Transitivity If A ⊆ B, B ⊆ C, then A ⊆ C Law of Contradiction (A ∩ A c ) = Φ De morgan laws (A ∪ B) c  = A c  ∩ B c (A ∩ B) c  = A c  ∪ B c

20. CONCLUSION: By using crisp sets,the set defined using characteristic function that assigns a boolean value.