Exercise 3.5.6 contains some false and incomplete claims, and therefore incomplete analysis

[hansonchar]

In Exercise 3.5.6, it says:

This statement is not true when $(A=\emptyset)\lor (B=\emptyset)$.

While this is true if one of $A,B$ is empty, the claim is false when both of them are empty. In fact, the statement

$$A\times B\subseteq C\times D\iff (A\subseteq C)\land(B\subseteq D)$$

holds vacuously if and only if both $A$ and $B$ are empty. Furthermore, when referring to the statement

$$A\times B=C\times D\iff(A=C)\land(B=D)$$

This statement is not true when $((A=\emptyset)\land(C=\emptyset))\lor ((B=\emptyset)\land(D=\emptyset))$.

While this is true, the claim is incomplete. A more complete analysis should lead to the fact that the statement holds vacuously if and only if $A,B,C,D$ are all empty. This suggests the current analysis of this exercise is incomplete.