Sets Representation & Its Types Notes | Study Mathematics (Maths) Class 11 - JEE

Introduction to Set

• We often deal with a group or a collection of objects, such as a collection of books, a collection of English alphabets, a class of students, a list of states in a country, a collection of coins, etc. 
• Set may be considered as a mathematical way of representing a collection or a group of objects.

Definition of Set

• “A set is a collection of well-defined objects.' In other words, a set is collection of defined and distinct objects. The objects of a set are called elements or members of the set. Further a set is always denoted by these brackets or braces { }.
  Example: A = is a set of the letters of the word “FOLLOW’’ 
  Then the set A will be a collection of the letters F, O, L, W.

Note: Here defined means existence of elements and distinct means there is no need to write repeated elements. 

• The main property of a set is that it is well defined. This means that given any object, it must be clear whether that object is a member (element) of the set or not.
  Generally, sets are named with the capital letters A, B, C, etc. 
• The elements of a set are denoted by the small letters a, b, c, etc. and if a is an element of a set A then a ‘belongs to’ the set A where the phrase ‘belongs to’ is denoted by ∈ (which is also known as epsilon). Then we write a ∈ A. If a is not the element of A then we write a ∉ A.
  Example: A is a set of natural numbers then 3 ∈ A and 0 ∉ A.  

Some examples of sets in daily life are:

• Students of our classroom.
• Collection of animals of the world.
• Collection of fruits.
• Collection of vegetables.

Which of the following collection are sets? Justify your answer.
Q.1. A collection of all-natural numbers less than 50.
Ans. It forms a set. The set will have the following elements.
A = {1, 2, 3, 4, 5, ................... 49}

Q.2. A collection of good hockey players in America
Ans. It doesn't form a set. Because the term "Good player" is vague and it is not well defined. However, a collection of players of hockey is a set.

Q.3. A collection of all girls in our class.
Ans. It forms a set. The elements of the above set are the names of those girls in the class.

Q.4. A collection of prime numbers less than 30
Ans. It forms a set. The set will have the following elements.
A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}

Q.5. A collection of ten most talented mathematics teachers.
Ans. It doesn't form a set. Because the term "Most talented" is vague and is not well defined. However, a collection of mathematics teachers is a set.

In this chapter we will have frequent interaction with some sets, so we reserve some letters for these sets as listed below:
N: The set of natural numbers
Z: The set of 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
C: The set of all complex numbers

➢ How to Represent a Set?

There are two methods to represent a set:

• Roster or tabular form.
• Set builder form.

1. Roster Form

In roster form, a set is described by listing the elements, separated by commas, within the braces.
Examples: 
(i) A set of vowels of English alphabets is described by {a, e, i, o, u}
(ii) A set of whole numbers is described by {1, 2, 3, 4, 5, 6,………..}
(iii) A set even natural numbers is described by {2, 4, 6, 8, 10,……..}

2. Set Builder Form
To understand set builder form we can take the help of some examples:
As discussed above, A = {a, e, i, o, u}
In set builder form, A = {x : x is a vowel of English alphabets} or {x | x is a vowel of English alphabets}
B = {1, 2, 3, 4, 5, 6,……….}
In set builder form, B = {x : x is a natural number} or {x | x is a natural number} or {x : x ∈ N} C = {2, 4, 6, 8, 10,……..}
In set builder form
C = {x : x = 2n, n ∈ N}

Note: The set of real numbers cannot be represented by roster form. In set builder form, A = {x : x ∈ R}

Different Kind of Sets

1. Empty Set or Null Set: A set containing no elements is called the empty set or null set or void set.
Reading Notation:

So, it is denoted by { } or ∅. 
Example: Consider the set A = {x : x < 1, x ∈ N}
There is no natural number which is less than 1. 
Therefore, A = { }, or ∅.

Note: The concept of empty set plays a key role in the study of sets just like the role of the number zero in the study of the number system.

► Cardinal Number of a Set (Order of a set) 
Cardinality or order of a set is defined as the number of elements that a set has. It is denoted by n(A) or c(A) or o(A) or |A| or  #A.
Example: A = {a, b, c} then n(A) = 3 and if A = { } or ∅ then n(A) = 0.

Note: The cardinal number of a finite set is finite.

3. Infinite Set: A set is said to be an infinite set if the number of elements in the set is not finite.
Example: Let W = The set of all whole numbers.
That is, W = {0, 1, 2, 3, ......................}
The set of all whole numbers contain infinite number of elements.
Hence, W is an infinite set. 

Note: The cardinal number of an infinite set is not a finite number.

Q.1. State, whether the given set is infinite or finite:
(i) {3, 5, 7, ….}
(ii) {1, 2, 3, 4}
(iii) {….., -3, -2, -1, 0, 1, 2}
(iv) {20, 30, 40, 50, ………., 200}
(v) {7, 14, 21, …………, 2401}
(vi) {0}
(vii) {∅}
(viii) {x | x is an even natural number less than 10, 000}
(ix) {All people in the world}
(x) {x | x is a prime number}
(xi) {x | x ∈ N and x > 10}
(xii) {….., -2, -1, 0}
Ans. Finite Sets: (ii), (iv), (v), (vi), (vii), (viii) & (ix)
Infinite Sets: (i), (iii), (x), (xi) & (xii)

Q.2. State, whether the following set is infinite or finite:
(i) Set of integers
(ii) {Multiples of 5}
(iii) {Fractions between 1 and 2}
(iv) {Numbers of people in India}
(v) Set of trees in the world
(vi) Set of prime numbers
(vii) Set of leaves on a tree
(viii) Set of children in all the schools of Delhi}
(ix) Set of seven natural numbers
(x) {……, -4, -, 2, 0, 2, 4, 6, 8}
(xi) {-12, -9, -6, -3, 0, 3, 6, ……}
(xii) {Number of points in a line segment 4 cm long}.
Ans. Finite Sets: (iv), (vii) & (viii)  
Infinite Sets: (i), (ii), (iii), (v), (vi), (ix), (x), (xi) & (xii)

Q.3. State, whether the following are finite sets, or infinite sets:
(i) {2, 4, 6, 8, …….., 800}
(ii) {-5, -4, -3, -2, …..}
(iii) {…., -5, -4, -3, -2,}
(iv) {Number of C.B.S.E. school in Delhi}
(v) {x : x is an integer between -60 and 60}
(vi) {No. of electrical appliances working in your house}
(vii) {x : x is a whole number greater than 20}
(viii) {x : x is a whole number less than 20}
Ans. Finite Sets: (i), (iv), (v), (vi) & (viii)
Infinite Sets: (ii), (iii) & (vii)

4. Singleton Set: A set containing only one element is called a singleton set. 
Example: Consider the set A = {x : x is an integer and 1 < x < 3}
So, A = {2}. That is, A has only one element.
Hence, A is a singleton set. 

Note: {0} is not null set. Because it contains one element. That is "0". 

5. Equivalent Set: Two sets A and B are said to be equivalent if they have the same number of elements.

In other words, A and B are equivalent if n(A) = n(B).

Reading notation

"A and B are equivalent" is written as A ≈ B

For example: 

Consider A = {1, 3, 5, 7, 9} and B = {a, e, i, o, u}

Here n(A) = n(B) =  5

Hence, A and B are equivalent sets.

6. Equal Sets: Two sets A and B are said to be equal if they contain exactly the same elements, regardless of order.

Otherwise, the sets are said to be unequal.

In other words, two sets A and B are said to be equal if

(i) Every element of A is also an element of B and

(ii) Every element of B is also an element of A.

Fig: A and B are Equal Sets

Reading notation:


Example: Consider A = {a, b, c, d} and B = {d, b, a, c}
Set A and set B contain exactly the same elements.
And also n(A) = n(B) = 4.
Hence, A and B are equal sets.
Note: If n(A) = n(B), then the two sets A and B need not be equal. Thus, equal sets are equivalent but equivalent sets need not be equal.

Q.1. State, if the given pairs of sets are equal sets or equivalent sets:
(a) {Natural numbers less than five} and {Letters of the word ‘BOAT’}
(b) {2, 4, 6, 8, 10} and {even natural numbers less than 12}
(c) {a, b, c, d} and {∆, ○, □, ∇}
(d) {Days of the week} and {Letters of the word ‘HONESTY}
(e) {1, 3, 5, 7, ……} and set of odd natural numbers.
(f) {Letters of the word ‘MEMBER’} and {Letters of the word ‘REMEMBER’}
(g) {Negative natural numbers} and {50th day of a month}
(h) {Even natural numbers} and {Odd natural numbers}
Ans. 
(a) Equivalent
(b) Equal
(c) Equivalent
(d) Equivalent
(e) Equal
(f) Equal
(g) Equal
(h) Equivalent