The Empty Set
Consider the set
A = { x : x is a student of Class XI presently studying in a school }
We can go to the school and count the number of students presently studying in Class XI 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 X and XI }
We observe that a student cannot study simultaneously in both Classes X and XI. Thus, the set B contains no element at all.
Definition 1 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 { }.
We give below a few examples of empty sets.
(i) Let A = {x : 1 < x < 2, x is a natural number}. Then A is the empty set, because there is no natural number between 1 and 2.
(ii) B = {x : x2 – 2 = 0 and x is rational number}. Then B is the empty set because the equation x2 – 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 the empty set, because 2 is the only even prime number.
(iv) D = { x : x2 = 4, x is odd }. Then D is the empty set, because the equation x2 = 4 is not satisfied by any odd value of x.
