Originally Posted by

**novice** My book says the converse of the following implication is false:

$\displaystyle A, B, C, D$ are sets such that $\displaystyle A\subseteq C$ and $\displaystyle B \subseteq D$, then $\displaystyle A \times B \subseteq C \times D$.

Although $\displaystyle P \Rightarrow Q \not \equiv Q \Rightarrow P$, I don't understand why since the above implication still can be true under special circumstances.

For exmple, When $\displaystyle A=D$ and $\displaystyle C=B $ or when $\displaystyle A = B = \emptyset $ or $\displaystyle C=D=\emptyset$ , I still can make it true.

I think $\displaystyle P \Rightarrow Q \not \equiv Q \Rightarrow Q$ is not always true. Am I missing something?