The Empty Set // Finite and Infinite Sets - Study Notes





1.3 The Empty Set


An Empty Set is a set that contains no elements.


Example


Let:


A = {x : x is a student of Class XI presently studying in a school}


We can go to the school and count the students studying in Class XI. Therefore, A contains a finite number of elements.


Now consider:


B = {x : x is a student presently studying in both Classes X and XI}


A student cannot study in Class X and Class XI simultaneously. Therefore, B contains no elements.


Hence,


B = { } = ∅


Here, ∅ represents the Empty Set or Null Set.


Note:



  • Empty Set: A set containing no elements.

  • Symbol: ∅ or { }

  • Also called: Null Set or Void Set


Point: A set having no element is called an Empty Set.


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1.3 The Empty Set


Definition 1: A set which does not contain any element is called the Empty Set, Null Set, or Void Set.


The Empty Set is denoted by: ∅ or { }


According to this definition:



  • B is an empty set because it contains no elements.

  • A is not an empty set because it contains elements.


Examples of Empty Sets


(i) Let
A = {x : 1 < x < 2, x is a natural number}


There is no natural number between 1 and 2.


Therefore, A = ∅.


(ii) Let
B = {x : x² − 2 = 0 and x is a rational number}

We have:


x² − 2 = 0
x² = 2
x = ±√2


Since √2 is irrational, there is no rational value of x satisfying the equation.


Therefore, B = ∅.


(iii) Let
C = {x : x is an even prime number greater than 2}

2 is the only even prime number. There is no even prime number greater than 2.


Therefore, C = ∅.


(iv) Let
D = {x : x² = 4, x is odd}

We have:


x² = 4
x = ±2


Both 2 and −2 are even, not odd.


Therefore, D = ∅.


Note:


A set containing no elements is called an Empty Set, Null Set, or Void Set.


Symbol: ∅ or { }






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1.4 Finite and Infinite Sets - Study Notes


1. Finite Set : A set which is empty or contains a definite number of elements is called a Finite Set.


Examples:


(i) Days of the week


W = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}


There are 7 elements.


Therefore, W is a finite set.


(ii) Solution of x² − 16 = 0


x² − 16 = 0
x² = 16
x = ± 4


So,


S = {−4, 4}


There are only 2 elements.


Therefore, S is a finite set.


2. Infinite Set : A set which does not contain a definite finite number of elements is called an Infinite Set.


Example:


Let G be the set of all points on a line. A line contains infinitely many points. Therefore, G is an infinite set.


3. Roster Form of Infinite Sets : In Roster Form, the elements of a set are written inside curly brackets { }. Since we cannot write all the elements of an infinite set, we write some elements and use three dots (...) to show that the pattern continues.


Examples


Natural Numbers: {1, 2, 3, 4, 5, ...}


Odd Natural Numbers: {1, 3, 5, 7, 9, ...}


Integers: {..., −3, −2, −1, 0, 1, 2, 3, ...}


All these are infinite sets.


Note:



  • Finite Set → Has a definite number of elements.

  • Infinite Set → Has infinitely many elements.

  • Empty Set → Has no elements.

  • Symbol of Empty Set → ∅ or { }


Three dots (...) → Show that the pattern continues.


Tip : i. Finite = Definite number,  ii. Infinite = No definite finite number,  iii. Empty = Zero elements


Finite and Infinite Sets — Study Notes


Not all infinite sets can be represented in Roster Form.


For example, the set of real numbers cannot be completely described in Roster Form because its elements do not follow a simple, particular pattern that allows us to list them.


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Example 6: Identify Finite and Infinite Sets


(i) {x : x ∈ N and (x − 1)(x − 2) = 0}


Solution:


(x − 1)(x − 2) = 0


Therefore,


x = 1 or x = 2


So, the set is: {1, 2}


It has a definite number of elements.


Ans: Finite Set


(ii) {x : x ∈ N and x² = 4}


Solution:


We have: x² = 4


Therefore,


x = ±2


But N = {1, 2, 3, ...}, so −2 is not a natural number.


Thus, the set is: {2}


It has only one element.


Ans: Finite Set


(iii) {x : x ∈ N and 2x − 1 = 0}


Solution:


We have:


2x − 1 = 0


2x = 1


x = 1/2


But 1/2 is not a natural number.


Therefore, the set contains no element: ∅


An empty set is considered a finite set because it has 0 elements.


Ans: Finite Set


(iv) {x : x ∈ N and x is prime}


Solution:


This is the set of all natural numbers that are prime: {2, 3, 5, 7, 11, 13, 17, ...}


There are infinitely many prime numbers.


Ans: Infinite Set


(v) {x : x ∈ N and x is odd}


This is the set of all odd natural numbers: {1, 3, 5, 7, 9, 11, ...}


There are infinitely many odd natural numbers.


Ans: Infinite Set


Note


Empty set is also a finite set.


Finite: Definite number of elements
Infinite: Infinitely many elements
Empty: 0 elements → Finite


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