# The Cartesian product of a set and the empty set

The cartesian product is the product of two sets. The resulting product is a set of pairs, one element belonging to $A$ and one element belonging to $B$ .

This can expressed as:

${A \times B = \{ (x,y) : x \in A, y \in B \} }$

So what happens if we try to take the cartesian product of a set and the empty set? ${A \times \emptyset}$ ?

We end up with: ${A \times \emptyset} = \emptyset$ . This is because we try to build a set like this: ${A \times B = \{ (x,y) : x \in A, y \in B \} }$ , but $B$ is empty.

It is impossible to create this resulting set of pairs because one of the sets is the empty set. As such, not a single element can be produced from $B$ , so our resulting set is the empty set.

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