Question

(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.

Answer 1

(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]