1.3 The Empty Set

NCERT Class 11 Mathematics Textbook for low vision students made screen readable by Professor T K Bansal.

Consider the set

A = { x : x is a student of Class 11 presently studying in a school }

We can go to the school and count the number of students presently studying in Class 11 in the school.

Thus, the set A contains a finite number of elements.

We now write another set B as follows:

B = { x : x is a student presently studying in both Classes 10 and 11 }

We observe that a student cannot study simultaneously both in Classes 10 and 11.

Thus, the set B contains no element at all.

Definition 1 A Null Set

'A set which does not contain any element is called the empty set, or the null set, or the void set.'

According to this definition, B is an empty set while A is not an empty set.

The empty set is denoted by the symbol φ, or by the { }.

We give below a few examples of empty sets.


(i) Let A = {x : 1 < x < 2, x is a natural number}. Then A is an empty set, because there is no natural number between 1 and 2.


(ii) B = {x : x^2 − 2 = 0 and x is a rational number}. Then B is an empty set because the equation


x^2 − 2 = 0 is not satisfied by any rational value of x.


(iii) C = {x : x is an even prime number greater than 2}.Then C is an empty set, because 2 is the only even prime number.


(iv) D = { x : x^2 = 4, x is odd }. Then D is an empty set, because the equation x^2 = 4 is not satisfied by any odd value of x.