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
$x \in C$ and $y \in D$
Thus, we have :
$A \subseteq C \& B \subseteq D$