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# Empty Set — Definition, Symbol, and Properties

TL;DR

The empty set is the one set that has no elements at all, written ∅ or {}, with cardinality 0. This article covers its symbol, why there is only one empty set, why it counts as a subset of every set, how it differs from 0 and from the number zero, and the mistakes that trip students up.

BT

**Last updated on June 29, 2022** 9 min read

## What Is The Empty Set?

The **empty set** is the unique set that contains no elements. It is written with the symbol ∅ or with empty curly braces {}, and its size — its **cardinality** — is 0. A set is any well-defined collection of objects, and the empty set is the collection you get when nothing meets the membership rule.

In set-builder form it looks like this:

∅ = { x : x ≠ x }

No object is unequal to itself, so the rule selects nothing. The result is a real set that happens to hold zero members.

### Why is there only one empty set?

There is exactly one empty set, which is why we say _the_ empty set, not _an_ empty set. Two sets are equal when they contain the same elements. Any two empty sets contain the same elements — namely, none — so they are equal. The set of living dinosaurs and the set of solutions to x² = −1 in the real numbers are the same set: ∅.

## Properties Of The Empty Set

The empty set obeys a small set of rules that every set problem leans on. Knowing them by name turns many questions into one-line answers.

- **Subset of every set:** ∅ ⊆ A for any set A. This holds vacuously, since there is no element of ∅ that could fail to be in A.

- **Union identity:** A ∪ ∅ = A. Uniting with the empty set adds nothing, so it acts like 0 does for addition.

- **Intersection with the empty set:** A ∩ ∅ = ∅. Nothing is shared with a set that holds nothing.

- **Cardinality zero:** |∅| = 0, which makes the empty set a _finite_ set.

- **Power set:** P(∅) = {∅}. The empty set has exactly one subset, itself, so its power set holds one element.

- **Cartesian product:** A × ∅ = ∅. Pairing every element of A with nothing produces no pairs.

## How The Empty Set Differs From Zero And From {0}

This is the single distinction that trips students most, so it earns its own line. Three objects look alike on the page and are not equal:

| Object                     | What it is                          | Cardinality            |
|---------------------------|-------------------------------------|------------------------|
| 0                         | a **number**                       | not a set             |
| ∅ (or {})                 | a **set with no elements**        | |∅| = 0         |
| {0}                       | a **set holding one element**, the number 0 | |{0}| = 1 |

The number 0 is not a set at all. The set {0} is a bag holding one thing, so it is _not_ empty — its cardinality is 1. Only ∅ is the empty bag. The set {0} is sometimes called the **zero set**, and reading it as "empty" because zero "feels like nothing" is exactly the trap to avoid.

## Why The Empty Set Earns Its Keep

Ask what the empty set is _for_, and the answer is consistency. Operations on sets must always return a set, even when they come up with nothing.

- Intersection has to land somewhere. When two sets share no members, A ∩ B needs a value. Without ∅, intersection would be undefined exactly when sets are disjoint, and disjoint sets are everywhere.

- It is the additive zero of set theory. Just as 0 leaves a number unchanged under addition, ∅ leaves a set unchanged under union: A ∪ ∅ = A. The parallel is not a coincidence; it is why the empty set behaves like a starting point.

- It builds the numbers themselves. In the standard construction of arithmetic, 0 is _defined_ as ∅, and 1 as {∅}, and so on. The whole number line grows out of nesting empty bags. That is the destination this small idea is pointing toward.

## Examples Of The Empty Set

### Example 1

**Write the set of all months with 32 days.**

No month has 32 days.

m : m is a month with 32 days = ∅.

The collection is well-defined; it just selects nothing. Final answer: ∅.

### Example 2 (where the first instinct goes wrong)

**Is {∅} the same as ∅?**

The first instinct is to read the braces as decoration and answer "yes — both are empty." Walk that out. If {∅} were empty, its cardinality would be 0. But count what is inside: there is one element, and that element is ∅ itself.

| {∅}| = 1, |∅| = 0

So they are not equal. The set {∅} is a bag containing one empty bag; ∅ is the empty bag. Final answer: **no — {∅} has one element, ∅ has none.**

### Example 3

**Find A ∩ B where A = {1, 2, 3} and B = {4, 5, 6}.**

The intersection holds elements common to both sets. A and B share nothing.

A ∩ B = ∅.

Two sets whose intersection is ∅ are called **disjoint**. Final answer: ∅.

### Example 4

**List every subset of ∅.**

A subset is a set whose every element also sits in the original. The only set with no elements to place is ∅ itself.

Subsets of ∅ = {∅}.

So the power set of the empty set is {∅}, a one-element set. Final answer: the only subset is ∅, and |P(∅)| = 1.

### Example 5

**Solve x² + 1 = 0 over the real numbers and write the solution set.**

Rearranging gives x² = −1. No real number squares to a negative.

x ∈ R : x² + 1 = 0 = ∅.

The equation has no real solution, so its solution set is empty. Final answer: ∅.

### Example 6

**A class has sets C = {students in chess club} and S = {students who row}. No student does both. Express the overlap, and find n(C ∪ S) given n(C) = 8 and n(S) = 5.**

The overlap is the intersection, and it is empty:

C ∩ S = ∅, n(C ∩ S) = 0.

Using the union count formula:

n(C ∪ S) = n(C) + n(S) - n(C ∩ S) = 8 + 5 - 0 = 13.

Final answer: 13.

## Where Students Trip Up On The Empty Set

### Mistake 1: Treating the empty set as the number zero

**Where it slips in:** When a problem mixes the symbol ∅, the set {0}, and the number 0 in the same line.

**Don't do this:** Writing ∅ = 0 or ∅ = {0}.

**The correct way:** ∅ is a set with no elements; 0 is a number; {0} is a set with one element (that element being 0). They are three different objects: |∅| = 0 but |{0}| = 1. The first instinct on {0} is to call it empty because zero "feels like nothing" — but the bag is holding something.

### Mistake 2: Thinking the empty set is not a subset

**Where it slips in:** Listing subsets of a set and leaving out ∅.

**Don't do this:** Claiming {1, 2} has only the subsets 1, 2, and {1, 2}.

**The correct way:** The empty set is a subset of _every_ set, so {1, 2} has four subsets: ∅, 1, 2, and {1, 2}. A set with n elements has 2ⁿ subsets, and the empty set is the one you forget. The student who skips ∅ here is usually the same one who later miscounts a power set.

### Mistake 3: Reading {∅} as empty

**Where it slips in:** Power-set problems and nested-set notation.

**Don't do this:** Writing |{∅}| = 0.

**The correct way:** {∅} contains one element — the empty set — so its cardinality is 1. The habit that fixes this is to count brace-pairs as bags: an outer bag holding one inner empty bag is not itself empty.

## The Mathematicians Behind the Empty Set

**Georg Cantor** (1845–1918, Germany) founded set theory in the 1870s, giving mathematics its first rigorous treatment of collections and their sizes, including sets with no members.

**André Weil** (1906–1998, France), a member of the Bourbaki group, introduced the now-standard ∅ symbol in 1939, borrowing it from the Norwegian alphabet.

## Conclusion

- The **empty set** is the unique set with no elements, written ∅ or {}, with cardinality 0.

- There is only one empty set, because any two sets with no elements are equal.

- The empty set is a subset of every set, which is why a set with n elements has 2ⁿ subsets.

- The most common mistake is confusing ∅ with the number 0 or with {0} — they are three distinct objects.

- The empty set anchors set operations and even the construction of the number 0 itself.

## Practice Questions on the Empty Set

Work through these:

1. Write the solution set of x² = −4 over the real numbers.

2. How many subsets does {a, b, c} have, and is ∅ one of them?

3. State the cardinality of {∅}.

4. Find A ∩ ∅ for A = {1, 2, 3}.

5. Are ∅ and {0} equal? Explain in one line.

**Answers**

1. ∅, since no real number squares to −4.

2. 8 subsets, and yes, ∅ is one of them.

3. |{∅}| = 1, since it holds one element, the empty set.

4. A ∩ ∅ = ∅ — nothing is shared with a set that holds nothing.

5. No. ∅ has 0 elements; {0} has 1 element (the number zero).