If $A \subseteq C$ and $B \subseteq D$ then $A \times B \subseteq C \times D$

Solution 1:

Here, it's a matter of spelling out that

• $A \subseteq C \iff a \in A \rightarrow a \in C$ and

• $B\subseteq D\iff b \in B \rightarrow b \in D$

• knowing that $A \times B = \{(a, b)\mid a\in A, b\in B\}$, and

• knowing that $C\times D = \{(c, d) \mid c \in C, d \in D\}$

Using the above: show that

$$A\times B \subseteq C \times D \;\;\text{ if and only if}\;\;\; (a, b) \in A\times B \rightarrow (a, b) \in C \times D$$

Solution 2:

Take any element of $A \times B$. It has the form $(a,b)$ where $a \in A$ and $b \in B$. We want to show that $(a,b) \in C \times D$. We have $a \in A$ and $A \subseteq C$, so $a \in C$. Perhaps you can take it from here.