# Crisp sets

24. Sep 2019
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### Crisp sets

• 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,……………….Xn} • 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/xϹA or XϹB}. • 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 εA and xεB}
• 16. Complement(c): • The complement of a set Ac(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 ∩ Ac) = Φ De morgan laws (A ∪ B)c = Ac ∩ Bc (A ∩ B)c = Ac ∪ Bc
• 20. CONCLUSION: • By using crisp sets,the set defined using characteristic function that assigns a boolean value.