The Empty Set

The Idea of a Set with No Elements

Most sets we think of contain something — the set of students in a class, the set of natural numbers less than 10, the set of letters in the word "mathematics". But what about a set that contains nothing at all? Can such a set exist?

Consider the set $A = \{ x : x \text{ is a student of Class XI presently studying in a school} \}$. You could walk into that school, count the students in Class XI, and get a finite number. $A$ has elements.

Now consider $B = \{ x : x \text{ is a student presently studying in both Classes X and XI} \}$. A single student cannot be enrolled in two different classes at the same time. No matter which school you go to, you will never find such a person. The set $B$ contains no element.

This is not a trick or a paradox — it is a perfectly valid set, and it has a special name.

Definition of the Empty Set

The empty set is denoted by the symbol $\phi$ (the Greek letter phi) or by the pair of braces with nothing inside: $\{ \}$.

In the example above, $B$ is an empty set, while $A$ is not.

Examples of Empty Sets

The textbook gives four clear examples. Work through each one carefully — they show the different ways a set can turn out to be empty.

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

Natural numbers are $1, 2, 3, 4, \dots$. There is no natural number that lies strictly between 1 and 2. The condition $1 < x < 2$ cannot be satisfied by any natural number. Therefore $A$ contains no elements — it is the empty set.

(ii) $B = \{ x : x^2 - 2 = 0 \text{ and } x \text{ is a rational number} \}$

The equation $x^2 - 2 = 0$ gives $x = \sqrt{2}$ or $x = -\sqrt{2}$. Both $\sqrt{2}$ and $-\sqrt{2}$ are irrational numbers — they cannot be expressed as a ratio of two integers. Since the condition requires $x$ to be rational, no value of $x$ satisfies both conditions. Hence $B$ is the empty set.

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

The only even prime number is 2. But the condition requires the number to be greater than 2. There is no even prime number greater than 2. So $C$ is the empty set. …