Question

If A × B ⊆ C × D and A × B ≠ ϕ, prove that A ⊆ C and B ⊆ D.

 

Answer 1

\[Let: \]
\[(x, y) \in (A \times B)\]
\[ \therefore x \in A, y \in B\]
\[Now, \]
\[ \because (A \times B) \subseteq (C \times D) \]
\[ \therefore (x, y) \in (C \times D) \]
\[Or, \]
\[x \in C \text{ and }y \in D\]
\[\text{ Thus, we have:} \]
\[ A \subseteq C  B \subseteq D\]