**Question:**

(i) If $A \subseteq B$, prove that $A \times C \subseteq B \times C$ for any set $C$.

(ii) If $A \subseteq B$ and $C \subseteq D$ then prove that $A \times C \subseteq B \times D$.

**Solution:**

(i) Given: A\subseteq B

Need to prove: $A \times C \subseteq B \times C$

Let us consider, $(x, y) \in(A \times C)$

That means, $x \in A$ and $y \in C$

Here given, $A \subseteq B$

That means, $x$ will surely be in the set $B$ as $A$ is the subset of $B$ and $x \in A$.

So, we can write $x^{\in} B$

Therefore, $x \in_{B}$ and $y \in_{C} \Rightarrow(x, y)^{\in}{ }_{(B \times C)}$

Hence, we can surely conclude that,

$A \times C \subseteq B \times C$ [Proved]

(ii) Given: $A \subseteq B$ and $C \subseteq D$

Need to prove: $A \times C \subseteq B \times D$

Let us consider, $(x, y) \in(A \times C)$

That means, $x \in_{\text {A and } y} \in_{C}$

Here given, $A \subseteq B$ and $C \subseteq D$

So, we can say, $x \in_{B}$ and $y \in_{D}$

$(x, y)^{\in}(B \times D)$

Therefore, we can say that, $A \times C \subseteq B \times D$ [Proved]