If A x B ⊆ C x D and A ∩ B ∈ ∅, Prove that A ⊆ C and B ⊆ D.
Given,
A × B ⊆ C x D and A ∩ B ∈ ∅
To prove : A ⊆ C and B ⊆ D
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.
A × B = {(a, b): a ∈ A and b ∈ B}
Since,
A × B ⊆ C x D (Given)
∴We can say (a, b) C × D
⇒ a ∈ C and b ∈ D
⇒ A ∈ C and B ∈ D
(A and B does not have common elements)
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.