When $A = \phi$, then number of elements in $P(A)$ is ______________.

When A = φ, then number of elements in P(A) is _______.

NCERT Exemplar · Q45

A = \phi

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Concept understanding — Empty Set

The Empty Set: Nothing but Something

Let's build this idea from the ground up.

The idea in plain words

Imagine you have a bag that is supposed to hold marbles.

If you put 3 red marbles in it, the bag contains a set of marbles.
If you put 0 marbles in it, the bag is empty.

That empty bag is not "nothing" — it is a real bag that exists, but it contains no objects.

In mathematics, the empty set is exactly that: a set that has no elements. It is a valid, well-defined set, just like the set of all planets in our solar system or the set of all even prime numbers greater than 2.

The key insight: The empty set is something — it is a set that contains nothing. That distinction is the whole game.

What it is and how to write it

Definition: The empty set (also called the null set) is the unique set that contains no elements.

You'll see it written in two ways:

$\emptyset$ (a slashed zero)
$\{ \}$ (a pair of curly braces with nothing inside)

Why this works — the properties you need

Here are the four things that make the empty set special. Memorise these for exams.

1. It has zero elements

The number of elements in the empty set is zero.

|\emptyset| = 0

2. It is a subset of every set

The empty set is a subset of any set $A$ .

\emptyset \subseteq A \quad \text{for every set } A

Why? Because there is no element in $\emptyset$ that is not in $A$ — the condition holds vacuously. Think of it like this: you can't find a counterexample if there's nothing to check.

3. There is only one empty set

$\emptyset$ and $\{ \}$ are the same set. There's no such thing as "two different empty sets."

4. It is NOT the same as $\{0\}$

This is where students slip. The set containing the number 0 is not empty — it has one element (the number 0). So:

\emptyset \neq \{0\}

Because $|\emptyset| = 0$ but $|\{0\}| = 1$ .

Set	What it contains	Number of elements
$\emptyset$	Nothing	0
$\{0\}$	The number 0	1
$\{ \emptyset \}$	The empty set itself	1

A common slip

Question: Is $\{ \emptyset \}$ the empty set?

Answer: No. $\{ \emptyset \}$ is a set that contains one element — that element is the empty set itself. So:

|\{ \emptyset \}| = 1

Think of it like a box that contains an empty bag. The box is not empty — it holds one thing (the empty bag).

Why does it matter? …

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The power set of the empty set is not empty — it contains exactly one element: the empty set itself. So the number of elements in $P(\phi)$ is $1$ .

The key here is to understand what the empty set really is, and what a power set does. Many students get tripped up because they think "empty" means "nothing", but in set theory, the empty set is a perfectly valid set — it's the set with no elements. And the power set of any set is the set of all its subsets.

Let's think about this intuitively. If you have a box with nothing in it, how many ways can you take things out of it? Exactly one way: you take nothing. That "taking nothing" corresponds to the empty set itself, which is always a subset of any set.

Now let's work through it step by step.

Recall the definition of power set.

For any set $A$ , the power set $P(A)$ is the set of all subsets of $A$ . That includes $A$ itself and the empty set $\phi$ , always.
Identify what $A$ is here.

Here $A = \phi$ , the empty set. So $A$ has 0 elements.

$n(A) = 0$ .
List all subsets of $\phi$ .

The only subset of the empty set is the empty set itself. Why? Because any subset must contain only elements from $A$ , and $A$ has no elements to choose from. So the only possible subset is $\phi$ .

So the subsets of $\phi$ are: $\{ \phi \}$ — that's one subset.
Write the power set.

$P(\phi) = \{ \phi \}$ . …

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Q.When A=ϕA = \phiA=ϕ, then number of elements in P(A)P(A)P(A) is ______________.

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Q.When $A = \phi$ , then number of elements in $P(A)$ is ______________.

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