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$?
Solution 1:
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.