# Why is the Cartesian product of $A$ and $\varnothing$ empty?

Given two sets $A$ and $\varnothing$, where $\varnothing$ is the empty set. The Cartesian product is defined by

$$A \times \varnothing = \{ (x,y) \mid x \in A, y \in \varnothing \}$$

Why is $A \times \varnothing = \varnothing$?

• the question is whether you can exhibit an element in $A\times \emptyset$? What might such an element look like... – Ittay Weiss Sep 19 '14 at 0:36

This is because for any $y$ in any universal set, $y \notin \emptyset$. It follows that even if $x \in A$, $(x,y) \notin A \times \emptyset$. The empty set has no elements.