Sets class 11

Higher Secondary Study Material
Plus 1 (Class 11) Mathematics
Chapter 1
Sets
A set is a well defined collection of distinct objects. The objects in a set are called elements. The number of elements in a set is called its cardinality or cardinal number. We shall denote sets by capital alphabets A,B,C ,…

State whether the following collection are sets? Justify the answer

1. The collection of all natural numbers less than 10

It is a set. It is a well defined collection

2. The collection of most talented writers of India

It is not a set. It is not a well defined collection

3. The collection of months of a year beginning with the letter J

It is a set. It is a well defined collection

4. A collection of most dangerous animals of the world

It is not a set. It is not a well defined collection

Methods to represent a set

1. Roster form(Tabular form)        2. Set builder fom

1.     ROSTER FORM

In this form, we are listing the elements, separated by commas and

enclose in braces {}. The elements can be written in any order.

ie, {1,2,3} and {2,3,1} are same

2.     SET BUILDER FORM

In this form, the characterizing property of the elements is stated.

For this, a variable x (denotes each element of the set) is written inside the braces,and then the symbol  ’ :  ‘ or ’  |  ‘and following it the common property of the elements is stated.

Eg:

Write the following set builder form into roster form

{x: x is a natural number less than 5}

Ans: {1,2,3,4}

{x : x is a letter in the word ’ BETTER’ }

Ans: {B,E,T,R}

{x : x is a prime number less than 10}

Ans: {2,3,5,7}

{x : x is an integer -1/2<x<  9/2}

Ans : {0,1,2,3,4}

{x : x is a two digit number such that the sum of its digits is 8}

Ans : {17,71,26,62,35,53,44,80}

N   : The set of all natural numbers W  : The set of all whole numbers Z : The set of all integers Z+   : The set of all positive integers Q   : The set of all rational numbers Q+   : The set of all positive rational numbers R : The set of all real numbers R+ : The set of all positive real numbers

Write the following set builder forms into roster forms

1. {x : x∈ N, 2< x ≤ 5}

2. {x : x∈Z, -2 ≤  x < 1}

3. {x : x∈N , x2 < 36}

4. {x: x is a letter in the word ‘TRIGONOMETRY’}

5. { x : x is an integer, 1/2 < x < 9/2 }

6. {x : x∈R, x2- 4 = 0 }

7. {x : x∈R, x2- 5x + 6 = 0}

8. { x : x is a positive divisor of 6}

9. {x : x is a prime divisor of 6}

10. {x : x∈Z, |x|<3}

Answers:

1. {3,4,5}

2. {-2,-1,0}

3. {1,2,3,4,5}

4. {T,R,I,G,O,N,M,E,Y}

5. -2 ,-1, 0, 0.5, 1 ,2 ,3, 4, 4.5 ,6   1/2  =0.5  and 9/2 = 4.5

 Ans : {1,2,3,4}

6. x2 – 4 = 0

 x2 = 4

  x  = √4

      = +2, -2

Ans: {-2,2}

7. x2-5x+6 = 0

It is a second degree equation

a =1, b = -5, c = 6

x = [−b ± √(b2 − 4ac)] / 2a

    = 2,3

Ans : {2,3}

8. {1,2,3,6}. (Divisor means the numbers which divides 6)

9. {2,3}        (1 and 6 are not prime)

10. {-2,-1,0,1,2} These are the integers whose absolute value is less than 3.

|-2|=2, |-1|= 1, |0| = 0, |1| =1, |2|= 2

∈ : Element of ( belongs to) ∉ : Not an element of (not belongs to)

The symbols ∈,∉ used to denote whether an object is an element of a set.

If a is an element of the set A, then we write a∈A

If a is not an element of the set A, then we write a∉A

Eg :

Consider A = {1,2,3,4}

1 ∈A  ,  2 ∈A , 0 ∉ A, 5 ∉ A

TYPES OF SETS

EMPTY SET :   If a set has no element, the set is called empty set or null set or void set. It is denoted by f or {}.  The cardinality of a null set is 0.

Eg : {x : x is a natural number between 2 and 3} is a null set.

     {x : x² + 1 = 0, x∈R} is a null set.

SINGLETON SET  :  A set consisting of a single element is called singleton set.

The cardinality of a singleton set is 1.

Eg :  {3} is a singleton set.

FINITE SET:  A set which is empty or having definite number of elements is called finite set. The number of elements of a finite set A is denoted by n(A).

Null set is a finite set.

Eg :

1. The set of natural numbers less than 100

2. The set of students in a school

INFINITE SET  : A set which is not finite is called infinite set.

Eg :

1. The set of all points in a line.

2. The set of real numbers.

INFINITE SET  : A set which is not finite is called infinite set.

Eg :

1. The set of all points in a line.

2. The set of real numbers.

3. The set of parallel lines in a plane.

EQUIVALENT SETS  : Two finite sets A and B are equivalent if they have same number of elements.

ie, n(A) = n(B)

Eg: 

 A = {1,2,3} and B = {a,b,c} are equivalent sets

EQUAL SETS  : Two sets A and B are  called equal sets if every element of A is an element of B and every element of B is an element of A(they have exactly the same elements). If A and B are equal we write A = B.

Eg: 

A = {1,2,3,4}, and B = {4,2,3,1}, then A = B.

SUBSETS AND SUPER SETS

⊂ : subset of (contained in) ⊃ : super set of

Consider two sets A and B. If every element of A is an element of B, then A is a subset of B and it is written as A⊂B   (read as A is a subset of B).

We also say that B is a super set of A and is written as B⊃A  (read as B is a super set of A).

Note

1.     If A ⊂B and B ⊂A, then A = B.

2.     Every set is a subset of itself. (A ⊂ A)

3.     Null set is a subset of every set.(f⊂A)

Eg: 

A= {1,3,4}, B = {1,2,3,4,5}.

Here every element of A is an element of B. A is a subset of B or B is a super set of A.

N⊂W⊂Z⊂Q⊂R

Write all possible subsets of the following sets

1. {1,2}

Ans:  {1},{2},{1,2}, {}

2. {3}

Ans: {3}, ⌽ 

3. {1,2,3}

Ans: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},⌽ 

4. {}

Ans: {}  (null set has only one subset)

The number of subsets of a set having n elements = 2n

Eg: 

The number of subsets of a set having 4 elements = 24 = 16

PROPER SUBSETS : The subsets of a set other than the set itself are called proper subsets

Eg:

*. Write the proper subsets of {1,2}

Ans: {1},{2},⌽ 

*.write the proper subsets of {-1,0,1}

Ans : {1-},{0},{1},{-1,0},{0,1},{-1,1},⌽ 

Proper subsets of a set having n elements = 2n-1 

*Write the number of proper subsets of a set having 4 elements.

Ans: 24-1 = 16-1 = 15.

POWER SET :  The set of all possible subsets of a given set A is called power set of A and is denoted by P(A). The number of elements in P(A) = 2n

Write the power set of the following sets

1. A = {1,2}

Ans: P(A) = { {1}, {2}, {1,2}, f}

2. B = {4}

Ans: P(B) = { {4}, f}

3. C = {2,3,4}

Ans : P(C) = { {2},{3},{4},{2,3},{3,4},{2,4},{2,3,4},f }

4. D = f

Ans :P(D)= {f}

5. Find the number of elements in power set of a set having 5 elements.

Ans: 2n = 25 = 32.

OPERATION ON SETS

UNION OF SETS : Let A and B be two sets. The union of A and B is the set of all elements of A and B and it is denoted by A⋃B(read as A union B). Every element of A⋃B belongs to A or to B or to A and B. 

Eg :

1.  A = {1,2,3}, B = {3,4,5}. Find A⋃B

Ans: A⋃B = {1,2,3}⋃{3,4,5} = {1,2,3,4,5}.

2.  P = {-1,1,2,3}, Q = {2,1,5} find P⋃Q

Ans: P⋃Q = {-1,1,2,3}⋃{2,1,5} = {-1,1,2,3,5}.

A ⋃ B = {x: x∈A or x∈B}

INTERSECTION OF SETS : Let A and B be two sets.The intersection of A and B is the set of elements which are common to A and B and it is denoted by A⋂B(read as A intersection B). Every element of A⋂B belongs to both A and B.  

Eg:

1.     A = {1,2,3,4}, B = {3,4,5,6}. Find A⋂B

Ans: A⋂B = {1,2,3,4}⋂{3,4,5,6} = {3,4}.

2.       P = {3,-1,4,5}, Q = {-1,4,7} Find P⋂Q.

Ans: P⋂Q = {3,-1,4,5}⋂{-1,4,7} = {-1,4}.

A⋂ B = {x: x∈A and  x∈B}

DISJOINT SETS : If A⋂B = f, A and B are called disjoint sets.

Eg : 

A = {1,2,3}, B = {4,5}

A⋂B = f. A and B are disjoint sets

Note:

If A ⊂ B, Then A⋃B = B and A⋂B = A

Eg: 

A = {2,3,4}, B = {1,2,3,4,5}

Here A is a subset of B.

A⋃B = {1,2,3,4,5} = B.

A⋂B = {2,3,4} = A.