Question

Show that if \({\bf{A}} \subseteq {\bf{C}}\)and \({\bf{B}} \subseteq {\bf{D}}\), then\({\bf{A \times B}} \subseteq {\bf{C \times D}}\).

Short answer

\(A \times B \subseteq C \times D\)

Step-by-step solution

Subset of a set

X is a subset of Y if every element of X is also an element of Y.

Therefore, \(X \subset Y\) means that X is a subset of Y, but X is not the same set of Y.

To prove \({\bf{A \times B}} \subseteq {\bf{C \times D}}\)

Proof:

First, A is a subset of C, \(A \subseteq C \Leftrightarrow \left\{ {x \in A \Rightarrow x \in C} \right\}\).

Similarly, B is a subset of D, \(B \subseteq D \Leftrightarrow \left\{ {x \in B \Rightarrow x \in D} \right\}\).

Now, to show that \(A \times B \subseteq C \times D\).

So, it is required to prove the implication \(\left( {y,z} \right) \in A \times B \Rightarrow \left( {y,z} \right) \in C \times D\).

Suppose that \(\left( {y,z} \right)\) belongs to \(A \times B\) i.e.,\(\left( {y,z} \right) \in A \times B\).

Which implies that y belongs to A for every z belongs to B.

i.e., \(\left\{ {y \in A} \right\}\Lambda \left\{ {z \in B} \right\}\)

Consequently,

Let \(\left\{ {y \in C} \right\}\Lambda \left\{ {z \in D} \right\}\), which means that y belongs to C for every z belongs to D.

The above holds true because \(A \subseteq C\;and\;B \subseteq D\).

Finally, \(\left( {y,z} \right) \in C \times D\) i.e., y and z belong to C X D.

Since, y and z are arbitrary it is proved that \(A \times B \subseteq C \times D\).

Therefore, it is proved that \(A \times B \subseteq C \times D\).