Answer
See solution and explanation below.
Work Step by Step
Let $A$, $B$, $C$, and $D$ be sets, and suppose $A\subseteq C$ and $B\subseteq D$. We want to show $A\times B \subseteq C \times D$. Recall the elements in $A \times B$ and $C \times D$ are ordered pairs, precisely, $A \times B = \{(a,b): a \in A \hspace{1 mm} \wedge \hspace{1mm} b \in B\}$ and $C \times D = \{(c,d): c \in C \hspace{1mm} \wedge \hspace{1mm} d\in D\}$. So to show $A \times B \subseteq C \times D$, let $(x,y) \in A \times B$. Then $x \in A$ and $y \in B$. Since $A \subseteq C$ and $B \subseteq D$, we have $x \in C$ and $y \in D$. Thus $(x,y) \in C \times D$. Therefore, $A \times B \subseteq C \times D$.
