**The Empty Set**

Definition:

Null set or Empty Set

Statement 1. A set which does not contain any element is called the empty set or the void set or null set and is denoted by {} or Ø.

Statement 2. A set which does not contain any element is called the empty set or the null set or the void set.

Statement 3. The set that contains no elements is called the empty set or null set and is symbolized by {} or Ø.

Statement 4. The set with no elements is called an empty set or null set. A Null set is designated by the symbol Ø.

Those 4 statements have the same meaning.

The null set is a subset of every set, i.e., If *A* is any set then Ø⊂*A*.

Note that {Ø} is not the empty set. This set contains the element Ø and has a cardi- nality of 1. The set {0} is also not the empty set because it contains the element 0. It also has a cardinality of 1.

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*:*x*^{2}-2=0 and *x* is rational number}. Then *B* is the 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 the empty set, because 2 is the only even prime number.

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

(v) The set of real roots of the polynomial *x*^{2}+9=0.

(vi) {*x*|5*x*=5*x*+2}.

Question 1. Which of the following sets are empty?

(i) *A*={*x*:*x*∈ℕ and *x*≤1}

(ii) *B*={*x*:3*x*+1=0, *x*∈ℕ}

(iii) *C*={*x*:2<*x*<3, *x*∈ℕ}

🌟 A set which does not contain any element is called the empty set.

Solution:

(i) *A*={*x*:*x*∈ℕ and *x*≤1}

Since, *x*≤1 i.e., *x*<1 and *x*=1, which is a natural number.

*A*={1}

So, this is not an empty set.

(ii)

*B*={

*x*:3

*x*+1=0,

*x*∈ℕ}

Since, 3

*x*+1=0

*x*=-⅓∉ℕ.

Thus, set

*B*does not contain any element. Hence, set

*B*is an empty set.

(iii)

*C*={

*x*:2<

*x*<3,

*x*∈ℕ}

Since, there is no natural number between 2 and 3. Hence, set

*C*is an empty set.

Question 2. Which of the following are examples of the null set?

(i) Set of odd natural numbers divisible by 2

(ii) Set of even prime numbers

(iii) {*x*:*x* is a natural numbers, *x*<5 and *x*>7}

(iv) {*y*:*y* is a point common to any two parallel lines}

Solution:

(i) A set of odd natural numbers divisible by 2 is a null set because no odd number is divisible by 2.

(ii) A set of even prime numbers is not a null set because 2 is an even prime number.

(iii) {*x*:*x* is a natural number, *x*<5 and *x*>7} is a null set because a number cannot be simultaneously less than 5 and greater than 7.

(iv) {*y*:*y* is a point common to any two parallel lines} is a null set because parallel lines do not intersect. Hence, they have no common point.

There is a special set that, although small, plays a big role. The empty set is the set {} that has no elements. We denote it as Ø so Ø={}.

Whenever you see the symbol Ø, it stands for {}. Observe that |Ø|=0. The empty set is the only set whose cardinality is zero.

Be careful in writing the empty set. Don’t write {Ø} when you mean Ø. These sets can’t be equal because Ø contains nothing while {Ø} contains one thing, namely the empty set. If this is confusing, think of a set as a box with things in it, so, for example, {2,4,6,8} is a “box” containing four numbers. The empty set Ø={} is an empty box. By contrast, {Ø} is a box with an empty box inside it. Obviously, there’s a difference: An empty box is not the same as a box with an empty box inside it. Thus Ø≠{Ø}. (You might also note |Ø|=0 and |{Ø}|=1 as additional evidence that Ø≠{Ø}.)

This box analogy can help us think about sets. The set *F*={ Ø,{Ø},{{Ø}}} may look strange but it is really very simple. Think of it as a box containing three things: an empty box, a box containing an empty box, and a box containing a box containing an empty box. Thus |*F*|=3. The set *G*={ℕ,ℤ} is a box containing two boxes, the box of natural numbers and the box of integers. Thus |*G*|=2.