(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 $A \times C \subseteq B \times D$.
(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]
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.