Empty Set Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Empty Set.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The empty set, denoted $\emptyset$ or $\{\}$ , is the unique set that contains no elements at all. It is a subset of every set because the statement 'every element of $\emptyset$ belongs to $A$ ' is vacuously true — there are no elements to contradict it.

Think of an empty box that is still a valid box—it just holds nothing. The empty set plays the same role for sets that zero plays for numbers: it is the identity element for union ( $A \cup \emptyset = A$ ) and the annihilator for intersection ( $A \cap \emptyset = \emptyset$ ). It is also a subset of every set, which keeps logical statements about 'all elements of $\emptyset$ ' vacuously true.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The empty set is the one set with zero elements, and it is a subset of every set.

Common stuck point: The procedure for empty set is the easy part; the trap is writing 'no answer' when a solution set is empty. Asking "Does this collection genuinely contain zero elements?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this collection genuinely contain zero elements?

Worked Examples

Example 1

easy

Determine whether each set is empty: (a)

\{x \in \mathbb{R} : x^2 = -1\}

, (b)

\{x \in \mathbb{Z} : 2 < x < 3\}

, (c)

\{0\}

Answer

(a)\;\emptyset,\quad (b)\;\emptyset,\quad (c)\;\text{not empty}

First step

(a)

x^2 = -1

has no real solution since squares are non-negative. This set is empty:

\emptyset

Full solution

2
(b) There is no integer strictly between 2 and 3. This set is empty: $\emptyset$ .
3
(c) $\{0\}$ contains the element $0$ . It is not empty; it has cardinality 1.

The empty set

\emptyset

contains no elements at all. A set containing zero (

\{0\}

) is not empty —

0

is a perfectly valid element.

Example 2

medium

Prove that the empty set

\emptyset

is a subset of every set

A

Example 3

medium

List all subsets of

\emptyset

Example 4

medium

Prove: for any set

A

A \setminus A = \emptyset

Example 5

hard

Prove:

\emptyset

is unique (no two empty sets can exist).

Example 6

hard

Find all

x \in \mathbb{R}

with

|x| < 0

Example 7

medium

Express the set of real solutions to

x^2 + 4 = 0

Example 8

challenge

Show that

\mathcal{P}(\mathcal{P}(\emptyset)) = \{\emptyset, \{\emptyset\}\}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Decide which are true: (a)

\emptyset = \{0\}

, (b)

\emptyset \subseteq \{1,2,3\}

, (c)

|\emptyset| = 0

Example 2

medium

Let

A = \emptyset

. Find: (a)

A \cup B

for any set

B

, (b)

A \cap B

for any set

B

, (c)

\mathcal{P}(A)

(the power set of

A

Example 3

easy

How many elements does the empty set

\emptyset

contain?

Example 4

easy

\emptyset = \{0\}

Example 5

easy

\emptyset

a subset of

\{1,2,3\}

Example 6

easy

How many elements does

\{\emptyset\}

have?

Example 7

easy

What is

A \cup \emptyset

for any set

A

Example 8

easy

What is

A \cap \emptyset

for any set

A

Example 9

easy

\emptyset \in \{\emptyset\}

Example 10

easy

\emptyset \subseteq \emptyset

Example 11

medium

How many subsets does the empty set have? List them.

Example 12

medium

Distinguish the cardinalities:

|\emptyset|

|\{\emptyset\}|

|\{\emptyset,\{\emptyset\}\}|

Example 13

medium

\{\} = \emptyset

? And is

\{\{\}\} = \emptyset

Example 14

medium

Why is the statement 'every element of

\emptyset

is even' true?

Example 15

medium

A \cap B = \emptyset

, what are

A

and

B

called, and can

A,B

both be nonempty?

Example 16

medium

Compute

|\mathcal{P}(\mathcal{P}(\emptyset))|

Example 17

medium

Solve

\{x \in \mathbb{R} : x^2 + 1 = 0\}

— what set is this?

Example 18

medium

A \subseteq \emptyset

, what must

A

be?

Example 19

medium

Simplify

(A \cup \emptyset) \cap \emptyset

Example 20

challenge

Prove that the empty set is unique (there is only one empty set).

Example 21

challenge

Using

A\cup\emptyset=A

and

A\cap\emptyset=\emptyset

, explain the analogy between

\emptyset

and the number 0.

Example 22

challenge

Show that

\emptyset

is the only set with no proper supersets among its own subsets — i.e.

\mathcal{P}(A)=\{A\}

iff

A=\emptyset

? Evaluate this claim.

Example 23

easy

\{\emptyset\} = \emptyset

Example 24

medium

\{x \in \mathbb{R} : x^2 + 1 = 0\}

empty?

Example 25

easy

\{x \in \mathbb{Z} : 0 < x < 1\}

empty?

Example 26

easy

True or false:

\emptyset \in \emptyset

Example 27

medium

Find

A \setminus B

A=\{1,2,3\}

and

B=A

Example 28

medium

A \cap B = \emptyset

, what term describes the sets

A

and

B

Example 29

medium

Let

A=\{x \in \mathbb{R}: x>5\}

and

B=\{x \in \mathbb{R}: x<3\}

. Find

A \cap B

Example 30

medium

True or false: 'every element of

\emptyset

is a horse' is true.

Example 31

medium

Determine whether

\{x \in \mathbb{N} : x+1=0\}

is empty.

Example 32

medium

A \subseteq \emptyset

, what is

A

Example 33

easy

True or false:

\emptyset \subseteq \{1,2\}

Example 34

hard

For sets

A,B

with

A \cup B = \emptyset

, what must be true of

A

and

B

← Back to Empty Set Formula Explained Practice Mode

Related Concepts

Set Cardinality

Background Knowledge

These ideas may be useful before you work through the harder examples.

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