MATHEMATICS XI SETS

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 { }.

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.

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}

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.

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.

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].

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

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

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,…………..}

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

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

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

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

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

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

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

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

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

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

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