Crisp sets are classical sets defined in boolean logic that have only two membership values - an element either fully belongs or does not belong to the set. Crisp sets are fundamental to the study of fuzzy sets. Key concepts of crisp sets include the universe of discourse, set operations like union and intersection, and properties like commutativity, associativity, distributivity and De Morgan's laws. Crisp sets provide a definitive yes or no for membership, unlike fuzzy sets which allow partial membership.

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.