1. MATHEMATICS XI
SETS

2. Sets: a well defined collection of distinct objects.
Objects, elements and members of a set are represented by small letters of English alphabet.
Sets are usually denoted by capital letters A,B,G,Q, etc.
The elements of a set are represented by small letters a, b, c, x, etc. each element of a set is denoted by curly brackets { }.

3. sets are represented by two methods:
Roaster or tabular form
Set builder form
Roaster form: in roaster form all the elements of a set are numerical values and they are separated by (,) and enclosed within curly brackets.
eg:- the set of natural numbers less than 6.
A= {1,2,3,4,5}
the set of vowels in English alphabet.
B= {a , e , I , o , u}
Note: in roaster form all the elements are taken as distinct.

4. Set builder form:- In set builder form each of the elements of a set possesses a single common property which is not possessed any element outside the set.
eg:- the set of vowels in the English alphabet
A= { x:x is a vowel in the English alphabet}
B= {x:x is a natural number and 3<x<=6}
C= {x:x is an odd natural number less than 3}

5. TYPES OF SETS
Equal sets: Two set A and B are said to be equal if they have exactly the same elements and we write A=B. they have exactly same elements.
Empty set: A set which doesn’t contain any element is called the empty set. It is denoted by { } or 0.
Finite and infinite set: A set which is empty or consists of a definite number of elements is called finite otherwise the set is called infinite.

6. Every set is subset of itself. i.e. ACB. 0 is subset of every set.
Subsets: A set A is said to be subsets of a set B if every element of A is also an element of B. it is represented as (ACB).
Important results:-
Every set is subset of itself. i.e. ACB.
0 is subset of every set.

7. Intervals of subsets of R
Open interval:- It is written in ( ).
eg:- {3,4,5,6,7,}  can be written as (2,8)
Closed interval:- In this interval, numbers are enclosed between [ ].
eg:- {3,4,5,6,7}  can be written as
[3,7]
Semi -closed interval:- The numbers are written between [ ).
eg:- {3,4,5,6,7} can be written as [3,8).
Semi- open interval:- The numbers are written between ( ].
eg:- {3,4,5,6,7} can be written as (2,7].

8. Power set and some important results:-
The collection of all the subsets of a set is called the power set of the given set.
Eg: A={1, 2, 3}
Subsets of A:- {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, 0.
Power set of A:- {{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, 0,{1,2,3,}}
If m is the no. of elements of a set A i.e. n(A)=m then, n(P(A))= 2m .
If m is the no. of elements of a set ‘A’ then, no. of proper sub-sets of ‘A’= 2m-1

9. N C W C I C Q C R, R-Q C R Where N:- natural numbers W:- whole numbers
Proper sub-set: let A and B be two sets if ACB and A=B. then, A is called a proper subset of B.
Super set: if set B contains the set A in it. Then, B is the super set of A.
Singleton set: it is a set containing only one element.
Subsets of a set of real numbers:-
N C W C I C Q C R, R-Q C R
Where N:- natural numbers
W:- whole numbers
I:- integers
Q:- rational numbers
R:- real numbers
R-Q:- irrational numbers

10. Universal set:- Universal set is the set which contains all sets in a given context is called universal set.
Eg:- A={1,2,3}
B={2,3,5}
c= {3,5,6,8,9}
Then, U = {1,2,3,5,6,8,9,…………..}

11. Venn diagrams
In Venn diagrams universal sets are represented by rectangles and sets by closed curves usually circles as shown in figure:- U

12. OPERATION OF SETS
Union of sets:- let A and B be any two sets. The union of A and B, is the set which consists of all the elements of A and all the elements of B, the common elements being taken only once. The symbol ‘U’ is used to denote the union.
AUB:- U

13. SOME PROPERTIES OF UNION OF SETS:-
Commutative law:- AUB=BUA
Associative law:- AU(BUC)= (AUB)UC
Law of identity element:- AUO= A, O is the identity.
Idempotent law:- AUA= A
Law of U:- UUA=U

14. Intersection of sets:- The intersection of sets A and B is the set of all elements which are common to both A and B. The symbol is used to denote the intersection.
A B:- U U U

15. Properties of intersection of sets:-
Commutative law:- A B
Associative property:- A (B C)= (A B) C
Law of O and U:- O A= O, U A=A.
Distributive law:- A (B C)=(A B) (A C) U U U U U U U U U U U U

16. A B= O then, A and B are said to be disjoint sets.
Disjoint sets:- if A and B are two sets such that
A B= O then, A and B are said to be disjoint sets.
A B:- U U U

17. Difference of sets:- The difference of sets A and B in t his order is the set of elements which belong to A but not to B.
A-B:-
B-A:- U U

18. Some important Venn diagrams:-U
BUC A B C U
A B U A B C

19. U A B
A (BUC) U C U A B
A C U C U A B
(A B) U (A C) U U C

20. PROPERTIES OF COMPLIMENT OF A SET:-
Compliment laws:- AUA’ = U
A A’ = O
Law of double complimentation:- (A’)’= A
Law of empty set and universal set:- O’ = U
U’= O U

21. De Morgan's law:-
The compliment of the union of two sets is the intersection of their compliments and the compliments of intersection of two sets is the union of their compliments.
(AUB)’= A’ B’
(A B)’= A’UB’ U U