Determining whether the given statements is true or not? $Problem$
Statement 1) $(A\subset C 
    ~\land~ B\subset D
) ~\to~ (A\times B)\subset (C\times D)$
Statement 2) $(A\times B)\subset (C\times D)~\to~(A\subset C 
    ~\land~  B\subset D
)$
Statement 3) Statement 2 if sets A and B are not empty
It is easy to see that statement 1 is true . I am confused about statement 2 and statement 3. 
Any suggestion will be appreciated. 
 A: *

*To avoid trivialities, let $A$ and $B$ be non-empty sets (the empty set is a subset of any set) such that $A \subset C$ and $B \subset D$. Suppose $(x,y) \in A \times B$. So $x \in A$ and $y \in B$, by definition of Cartesian product. Since $A \subset C \land B \subset D$, we have that $x \in C$ and $y \in D $. This means that $(x,y) \in C \times D$, and so $(A \times B) \subset (C \times D).$

*Consider $A=\{0,1\}$, $B= \emptyset$, $C=\{0\}$, and $D=\{1\}$. Since $A \times \emptyset = \emptyset$ and $\emptyset \subset \{(0,1)\}=C \times D$, we have that $(A \times B) \subset (C \times D)$ but clearly $\neg(A \subset C)$ ($1 \in A$ but $1 \notin C$) .

*This is true and I will leave it to you (1 and 2 seem like a good hint).


Note: The issue with 2 is that we may possibly have $A=\emptyset$ or $B=\emptyset$. If either of these cases is true, then $A \times B = \emptyset$ and so $(A \times B) \subset (C \times D)$ for any sets $C$ and $D$. In other words, the issue with not stating the assumption "$A$ and $B$ are non-empty" is that we might suppose $(x,y) \in A \times B$ when no such pair of elements even exists.
