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.

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

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 RWhere 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

UU

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

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:-

A B

C

U

A B

C

U

BUC

A B

U

A B

C

U

A B

C

U

A B

C

U

A (BUC)

U(A B) U (A C)

A CU

UU

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