What happens if you do a cartesian product of a set by the empty set?

Question

Lets say you do A x B with A being an arbitrary set and B being the empty set. How could a cartesian point actually be constructed...

Answer 1

For all sets $A$ one has that $A\times \emptyset = \emptyset = \emptyset \times A$.

This is seen directly as a result of the definition of cartesian product: $A\times B = \{(a,b)~:~a\in A,b\in B\}$ . In the case of $A\times \emptyset$ this is the set of all ordered pairs $(a,b)$ where $a\in A$ and $b\in \emptyset$. Since there are no elements in the empty set, there are no such valid choices for $b$ hence no valid ordered pairs $(a,b)$ to reside in the cartesian product.

Note further the multiplication rule: $|A\times B| = |A|\times |B|$.

You have that $|A\times \emptyset| = |A|\times |\emptyset| = |A|\times 0 = 0$ and the only set with zero elements is the empty set.

Answer 2

Suppose $A\times B\neq\varnothing$. Hence, by AoC, there exists an element $(a,b)\in A\times B$, so, by definition, $a\in A$ and $b\in B$, but this is not true, because B is empty, so $A\times B$ is empty.