Is it true that the empty set is contained in every Cartesian Product? Is it true to say that $\forall A,B$ (set)
$\emptyset \subseteq A\times B $
Or: The empty set is contained in every Cartesian Product? (As it is contained in every set).
 A: If the product of two sets is a set then the answer is yes.
Now it depends on how you were taught:


*

*It could be that you were taught naively, and then you were just given the axiom that the product of two sets is a set. Maybe it wasn't even called an axiom, just a definition which said that $A\times B$ is a set.

*It could be that you were taught axiomatically and were given the axiom of pairing, i.e. for every two elements $x,y$ there is an ordered pair $\langle x,y\rangle$ and then you construct $A\times B$ using the replacement axiom, $A\times B=\{\langle a,b\rangle\mid a\in A\land b\in B\}$.

*It could be that you were taught that $\langle a,b\rangle=\{\{a\},\{a,b\}\}$ (or some other encoding of ordered pairs via sets), and then using subset axiom (or replacement, again) you can deduce that $A\times B\subseteq\mathcal P(\mathcal P(A\cup B))$ (in the case given here, but you may have $\mathcal P(A\cup\mathcal P(B))$ instead if you encode ordered pairs differently).
In all these cases the Cartesian product of two sets is a set, and therefore $\varnothing$ is a subset of this product. Why is this point so important? Because we define $\subseteq$ as a relation between sets, and then we can extend it to larger collections by one trick or another.
Of course you can ask yourself generally whether or not the definition of $\subseteq$ holds, i.e. every element of the empty set is an element of $A\times B$. Of course this holds vacuously, there is no counterexample.  But then there is a delicate issue about how to apply the definition of $\subseteq$ to non-sets. But once you know that $A\times B$ is a collection of objects from your universe it should suffice to conclude that $\varnothing\subseteq A\times B$ is a legitimate and meaningful sentence.
A: For every set $S$ one has $\emptyset\subseteq S$.
Given sets $A$ and $B$, let $S=A\times B$.
$S$ is a set, so ...
A: [Following the suggestion from @Babak Sorouh]
Are you able to find an element of the empty set which is not an element of the product?
This is the reason!
