The document discusses classical or crisp set theory. Some key points: 1) Classical set theory deals with sets that have definite membership - an element either fully belongs to a set or not. This is represented by true/false or yes/no. 2) A set is a well-defined collection of objects. The universal set is the overall context within which sets are defined. 3) Set operations like union, intersection, complement and difference are used to combine or relate sets according to specific rules. 4) Properties like commutativity, associativity and distributivity define the logical behavior of sets under different operations.

1. 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,………an}
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. 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. 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