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Question
If A × B ⊆ C × D and A × B ≠ ϕ, prove that A ⊆ C and B ⊆ D.
Solution
\[Let: \]
\[(x, y) \in (A \times B)\]
\[ \therefore x \in A, y \in B\]
\[Now, \]
\[ \because (A \times B) \subseteq (C \times D) \]
\[ \therefore (x, y) \in (C \times D) \]
\[Or, \]
\[x \in C \text{ and }y \in D\]
\[\text{ Thus, we have:} \]
\[ A \subseteq C B \subseteq D\]
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