Empty Set — Definition, Symbol, and Properties

What Is The Empty Set?

The empty set is the unique set that contains no elements. It is written with the symbol $\varnothing$ 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:

$$\varnothing = {, x : x \neq x ,}$$

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 $x^2 = -1$ in the real numbers are the same set: $\varnothing$.

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: $\varnothing \subseteq A$ for any set $A$. This holds vacuously, since there is no element of $\varnothing$ that could fail to be in $A$.

• Union identity: $A \cup \varnothing = A$. Uniting with the empty set adds nothing, so it acts like $0$ does for addition.

• Intersection with the empty set: $A \cap \varnothing = \varnothing$. Nothing is shared with a set that holds nothing.

• Cardinality zero: $\lvert \varnothing \rvert = 0$, which makes the empty set a finite set.

• Power set: $P(\varnothing) = {\varnothing}$. The empty set has exactly one subset, itself, so its power set holds one element.

• Cartesian product: $A \times \varnothing = \varnothing$. 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:

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 $\varnothing$ 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, $A \cap B$ needs a value. Without $\varnothing$, 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, $\varnothing$ leaves a set unchanged under union: $A \cup \varnothing = A$. 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 $\varnothing$, and $1$ as ${\varnothing}$, and so on. The whole number line grows out of nesting empty bags. That is the destination this small idea is pointing toward.

Real databases run on the same logic. A search that matches no records returns an empty result set, not an error — and code that forgets this is where bugs live.

Examples Of The Empty Set

Example 1

Write the set of all months with 32 days.

No month has 32 days.

$${m : m \text{ is a month with 32 days}} = \varnothing$$

The collection is well-defined; it just selects nothing. Final answer: $\varnothing$.

Example 2 (where the first instinct goes wrong)

Is ${\varnothing}$ the same as $\varnothing$?

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

$$\lvert {\varnothing} \rvert = 1, \qquad \lvert \varnothing \rvert = 0$$

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

Example 3

Find $A \cap B$ where $A = {1, 2, 3}$ and $B = {4, 5, 6}$.

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

$$A \cap B = \varnothing$$

Two sets whose intersection is $\varnothing$ are called disjoint. Final answer: $\varnothing$.

Example 4

List every subset of $\varnothing$.

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

$$\text{Subsets of } \varnothing = {\varnothing}$$

So the power set of the empty set is ${\varnothing}$, a one-element set. Final answer: the only subset is $\varnothing$, and $\lvert P(\varnothing) \rvert = 1$.

Example 5

Solve $x^2 + 1 = 0$ over the real numbers and write the solution set.

Rearranging gives $x^2 = -1$. No real number squares to a negative.

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

The equation has no real solution, so its solution set is empty. (Over the complex numbers the story changes — a phantom of the imaginary unit $i$ waiting in a later chapter.) Final answer: $\varnothing$.

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 $n(C \cup S)$ given $n(C) = 8$ and $n(S) = 5$.

The overlap is the intersection, and it is empty:

$$C \cap S = \varnothing, \qquad n(C \cap S) = 0$$

Using the union count formula:

$$n(C \cup S) = n(C) + n(S) - n(C \cap S)$$

$$n(C \cup S) = 8 + 5 - 0 = 13$$

Because the sets are disjoint, the union is just the two sizes added. 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 $\varnothing$, the set ${0}$, and the number $0$ in the same line.

Don't do this: Writing $\varnothing = 0$ or $\varnothing = {0}$.

The correct way: $\varnothing$ 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: $\lvert \varnothing \rvert = 0$ but $\lvert {0} \rvert = 1$. 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 $\varnothing$.

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: $\varnothing$, ${1}$, ${2}$, and ${1, 2}$. A set with $n$ elements always has $2^n$ subsets, and the empty set is the one you forget. The student who skips $\varnothing$ here is usually the same one who later miscounts a power set.

Mistake 3: Reading ${\varnothing}$ as empty

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

Don't do this: Writing $\lvert {\varnothing} \rvert = 0$.

The correct way: ${\varnothing}$ 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 $\varnothing$ symbol in 1939, borrowing it from the Norwegian alphabet.

Conclusion

• The empty set is the unique set with no elements, written $\varnothing$ 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 an $n$-element set has $2^n$ subsets.

• The most common mistake is confusing $\varnothing$ 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, then check your answers below.

1. Write the solution set of $x^2 = -4$ over the real numbers.

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

3. State the cardinality of ${\varnothing}$.

4. Find $A \cap \varnothing$ for $A = {1, 2, 3}$.

5. Are $\varnothing$ and ${0}$ equal? Explain in one line.

Answers

1. $\varnothing$, since no real number squares to $-4$.

2. $2^3 = 8$ subsets, and yes, $\varnothing$ is one of them.

3. $\lvert {\varnothing} \rvert = 1$, since it holds one element, the empty set.

4. $A \cap \varnothing = \varnothing$ — nothing is shared with a set that holds nothing.

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

A Practical Next Step

Practice these to solidify your understanding: list all subsets of ${a, b, c}$ and confirm there are $2^3 = 8$, including $\varnothing$; then decide whether ${\varnothing}$, $\varnothing$, and ${0}$ are equal in any pairing. If you get stuck on the subset count, come back to Example 4 and the second mistake above.

To go further, read how the empty set sits inside the broader rules in sets, how it shows up in intersection of sets, why it counts among subsets, how its size relates to cardinality, and how it is drawn in a Venn diagram. Want a live Bhanzu trainer to walk through more empty set problems? Book a free demo class.