EMPTY SET
A Set which usually does not contain any members is to be called an “Empty Set” or a “Null Set” and the empty set is denoted or implies by φ or { }
For Example –
1) The Set { x/ 5x + 4 = 0, x ϵ W } is to be considered as an empty set because,
5X + 4 = 0,
=> 5x = -4
=> x = - 4/5 ∉ A.
2) The Set { x/6x + 15 = 12, x ϵ W } is to be considered as an empty set because,
6x + 15 = 12
=> 6x = 12 - 15
=> 6x = -3
=> x = - 3/6 = - 1/2 ∉ B
Example – 1
Identify the Empty Set, Singleton Set & Pair Set
1) A = { x / x + 5 = 4, x ϵ A }
=> X = -5 + 4 = - 1 , -1 ϵ A and A = { -1 }, So n(A) = 1
So, set 'A' is Singleton set.
Example – 2
Z = { x / x² + 5 = 2, x ϵ Z }
Ans.) x² + 5 = 2 ;
x² = 2 - 5 = - 3, for any natural numbers x, So the set has no members, n(Z) = 0
Therefore set ‘Z’ is an empty set.
Example – 3
M = { x / x² + 2 = 0, x ϵ Z }
Ans.) x² + 2 ≠ 0, for any natural number x, so the set ‘M’ has no members, n(M) = 0.
n(M) = 0, therefore ‘M’ is an empty set.
Example – 4
B = { x / x² = 25, x ϵ Y }
Ans.) x² = 25
=> x = 5 , 5 & - 5 ϵ Y.
So, Y = { 5, -5 } , then n (Y) = 2
So, ‘Y’ is a pair set.