What Is the Cartesian Product with the Empty Set?

For any set $A$, the Cartesian product with the empty set is empty in either order:

$$A\times\varnothing=\varnothing\times A=\varnothing.$$

This follows directly from the definition of a Cartesian product: each member must be an ordered pair whose first coordinate belongs to one set and whose second coordinate belongs to the other. Since $\varnothing$ has no elements, no such pair can be formed.

The definition of a Cartesian product

For sets $A$ and $B$, their Cartesian product is

$$A\times B=\{(a,b):a\in A\text{ and }b\in B\}.$$

The notation $(a,b)$ means an ordered pair: its first coordinate is $a$ and its second coordinate is $b$. To be an element of $A\times B$, a pair must satisfy both membership conditions at once: $a\in A$ and $b\in B$.

When $B=\varnothing$, the second condition would require an element $b$ of the empty set. There is no such $b$. Therefore, no ordered pair meets the definition, regardless of what $A$ is. The same reasoning applies when the first factor is empty.

A short proof

Suppose, for a contradiction, that some pair $(a,b)$ belongs to $A\times\varnothing$. By the definition of the product, this would imply $a\in A$ and $b\in\varnothing$. But $\varnothing$ has no elements, so $b\in\varnothing$ is impossible. Thus $A\times\varnothing$ has no elements and equals $\varnothing$.

For the other order, if $(a,b)\in\varnothing\times A$, then the definition would require $a\in\varnothing$, which is impossible. Hence $\varnothing\times A=\varnothing$ as well.

Worked examples

Let $A=\{1,2,3\}$. Then

$$A\times\varnothing=\varnothing.$$

You might try to list pairs by choosing a first coordinate from $A$ and a second from $\varnothing$. You can choose $1$, $2$, or $3$ for the first coordinate, but there is no possible second coordinate. The list of pairs is therefore empty. Reversing the factors gives the same result:

$$\varnothing\times A=\varnothing.$$

If both factors are empty, the rule still applies:

$$\varnothing\times\varnothing=\varnothing.$$

No pair can be formed because neither coordinate can be chosen from its factor.

Why $(\varnothing,\varnothing)$ is not in the product

A common confusion is to treat the empty set itself as though it were an element available for selection. It is a set, but it is not an element of itself: $\varnothing\notin\varnothing$. The product $\varnothing\times\varnothing$ requires both coordinates to be elements of $\varnothing$, so the pair $(\varnothing,\varnothing)$ does not qualify.

However, the singleton set $\{\varnothing\}$ is not empty. It has one element, namely $\varnothing$. This changes the product:

$$\{\varnothing\}\times\{\varnothing\} =\{(\varnothing,\varnothing)\}.$$

The braces matter: $\varnothing$ has no elements, while $\{\varnothing\}$ has one.

Connection to cardinality

For finite sets, the number of elements in a Cartesian product is the product of the numbers of elements in its factors:

$$|A\times B|=|A|\,|B|.$$

Since $|\varnothing|=0$, this gives $|A\times\varnothing|=|A|\cdot 0=0$, consistent with the definition. The definition is the more fundamental reason: there are no ordered pairs when a coordinate would have to come from an empty set.

Common mistakes

• Confusing $\varnothing$ with $\{\varnothing\}$. The empty set has zero elements; the singleton containing it has one.
• Treating the empty set as an available coordinate. A coordinate must be an element of its factor. The fact that $\varnothing$ is itself a set does not make it an element of $\varnothing$.
• Thinking order changes the conclusion. In general, $A\times B$ and $B\times A$ need not be the same set, because ordered pairs distinguish their coordinates. But if either factor is empty, both products have no elements.
• Adding an “empty pair.” The empty product is the set with no ordered pairs, not a set containing a special pair.