Given: A × B ⊆ C × D and A × B ≠ ϕ

Need to prove: A ⊆ C and B ⊆ D

Let us consider, (x, y) (A × B) ---- (1)

⇒ (x, y) (C × D) [as A × B ⊆ C × D] ---- (2)

From (1) we can say that,

x A and y B ---- (a)

From (2) we can say that,

x C and y D ---- (b)

Comparing (a) and (b) we can say that,

⇒ x A and x C

⇒ A ⊆ C

Again,

⇒ y B and y D

⇒ B ⊆ D [Proved]