Question

If A x B ⊆ C x D and A ∩ B ∈ ∅, Prove that A ⊆ C and B ⊆ D.

Answer 1

Given as

Here, A × B ⊆ C x D and A ∩ B ∈ ∅

A × B ⊆ C x D denotes A × B is subset of C × D that is every element A × B is in C × D.

And A ∩ B ∈ ∅ denotes A and B does not have any common element between them.

Now, A × B = {(a, b): a ∈ A and b ∈ B}

∴We can say that (a, b) ⊆ C × D [Since, A × B ⊆ C x D is given]

a ∈ C and b ∈ D

a ∈ A = a ∈ C

A ⊆ C

And

b ∈ B = b ∈ D

B ⊆ D

Thus proved.