1. SET THEORY
Chumki Sarkar

2. Definition:- A collection of objects is defined to be a set when
(i) the collection is well-defined;
(ii) objects belonging to the collection are different;
(iii) objects of the collection are independent of the
order of their arrangement
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

3. METHODS OF SET PRESENTATION
Presentation of Set
ROSTER or TABULAR METHOD
PROPERTY or SET-BUILDER METHOD
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

4. ROSTER METHOD
The set is described by just listing its elements inside brackets.
Ex:the set of all vowels of the English alphabet A={a,e,i,o,u}
Demerits:This method fails if all the elements of the set cannot be displayed.In that case the property method is used.
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

5. PROPERTY OR SET-BUILDER METHOD
The set is described as A={x/P(x)} where x is any element possessing the property P(x).
Ex:The set of all prime numbers is described as A={x/x is a prime number}
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

6. SINGLETON SET
A set consisting of only one element is called singleton set.
Ex:A={4}
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

7. NULL SET
A set which contains no elements at all is called null set or empty set or void set.It is denoted by Φ( Phi).
Ex:Φ={x:x is an integer and 2<x<3}
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

8. FINITE AND INFINITE SETS
A set is said to be finite if the number of elements contained in the set can be counted and the counting process has an end.
Ex: A={10,11,12,12,13,14,15} i.e. n(A)=6
A set is said to be infinite if the number of elements contained in the set can not be determined by counting.
Ex: A={1, 1/2 ,1/3 , 1/4 , }
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

9. EQUAL SET
Two sets A & B are said to be equal if all the elements of A are the elements of B as well as all the elements of B are the elements of A and written as A=B
Example: A={1,2,3,4}
B={2,4,1,3} then A=B
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

10. UNIVERSAL SET
A set U is called Universal set if all the sets under consideration are the subsets of U.
Example: U={1,2,3,4,5,6,7,8,9} :Universal Set
A={2,4,6,8} :Subset 5 U 1 7 4 2 A 3 8 6 9
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

11. SUB SET U A B A = {3, 9}, B = {5, 9, 1, 3}, A  B
A  B “A is a subset of B”
A  B if and only if every element of A is also an element of B.
Examples:
A = {3, 9}, B = {5, 9, 1, 3}, A  B U A B
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

12. DISJOINT SET
Two sets A & B are said to be disjoint if they have no common elements. Example: A={1,2,3} & B={4,5,6} U A B
Chumki Sarkar
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

13. UNION OF TWO SETS
The Union of two given sets A & B written as A ⋃ B is defined to be the set of all elements which belong either to A or to B or to both.
Example: A={1,2,3,4} & B={3,4,5,6}
then, A ⋃ B = {1,2,3,4,5,6} U A B
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

14. INTERSECTION OF TWO SETS
The intersection of two sets A & B written as A ⋂ B is the set of all elements which belong to both A & B
Example: A={1,2,3,4} & B={3,4,5,6}
then, A ⋂ B = {3,4,} U A B
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

15. DIFFERENCE OF TWO SETS U A B
The difference between two sets A & B written as A-B is the set of all elements which belong to A but which do not belong to B
Example: A={1,2,3,4} & B={3,4,5,6}
then, A - B = {1,2} U A B
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

16. COMPLEMENT OF A SET The compliment of a set A written as
is the set of all elements of the universal set U which do not belong to A
Example: A={1,2,3,4} & U= {1,2,3,4,5,6,7,8}
then, = {5,6,7,8} U A
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

17. LAWS OF SET OPERATIONS Associative Laws
Commutative Laws
i) A⋃B = B⋃A,
ii) A⋂B = B⋂A
for any two sets A & B
Associative Laws
i) A⋃(B⋃C)=(A⋃B)⋃C, ii) A⋂(B ⋂C)=(A⋂B)⋂C
for any 3 sets A,B & C
Distributive Laws
i) A⋃(B⋂C)=(A⋃B)⋂(A⋃C), ii) A⋂(B⋃C)=(A⋂B)⋃(A⋂C)
for any 3 sets A,B & C
De Morgan’s Laws
(A⋃B)C =AC⋂BC
ii) (A⋂B)C=AC⋃BC
for any two sets A & B
Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

18. Commutative Laws = = Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

19. Associative Laws (i) Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

20. Associative Laws (ii) Chumki Sarkar From equation (i) & (ii)
Presented by: Chumki Sarkar, Maheshtala College

21. Associative Laws Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

22. Associative Laws Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

23. Distributive Laws Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

24. Distributive Laws Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

25. Distributive Laws Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

26. Distributive Laws Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

27. De Morgan’s Laws Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

28. De Morgan’s Laws Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College

29. THANK YOU Chumki Sarkar
Presented by: Chumki Sarkar, Maheshtala College