(i) If A ⊆ B, prove that A × C ⊆ B × C for any set C.

(ii) If A ⊆ B and C ⊆ D then prove that A × C ⊆ B × D.

(i) Given: A ⊆ B

Need to prove: A × C ⊆ B × C

Let us consider, (x, y) (A × C)

That means, x A and y C

Here given, A ⊆ B

That means, x will surely be in the set B as A is the subset of B and x A.

So, we can write x B

Therefore, x B and y C ⇒ (x, y) (B × C)

Hence, we can surely conclude that,

A × C ⊆ B × C [Proved]

(ii) Given: A ⊆ B and C ⊆ D

Need to prove: A × C ⊆ B × D

Let us consider, (x, y) (A × C)

That means, x A and y C

Here given, A ⊆ B and C ⊆ D

So, we can say, x B and y D

(x, y) (B × D)

Therefore, we can say that, A × C ⊆ B × D [Proved]