1.

For any sets A and B, prove that(A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A)

Answer»

Given: A and B two sets are given. 

Need to prove: (A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A) 

Let us consider, (x, y)∈(A × B) ∩ (B × A) 

⇒ (x, y)∈(A × B) and (x, y)∈(B × A) 

⇒ (x∈A and y∈B) and (x∈B and y∈A)

⇒ (x∈A and x∈B) and (y∈B and y∈A) 

⇒ x∈(A ×B) and y∈(B × A)

⇒ (x, y)∈(A × B) ∩ (B × A) 

From this, we can conclude that, 

⇒ (A × B) ∩ (B × A) ⊆ (A ∩ B) × (B ∩ A)..... (1) 

Let us consider again, (a, b)∈(A ∩ B) × (B ∩ A) 

⇒ a∈(A ∩ B) and b∈(B ∩ A) 

⇒ (a∈A and a∈B) and (b∈B and b∈A) 

⇒ (a∈A and b∈B) and (a∈B and b∈A) 

⇒ (a, b)∈(A × B) and (a, b)∈(B × A) 

⇒ (a, b)∈(A × B) ∩ (B × A) 

From this, we can conclude that, 

⇒ (A ∩ B) × (B ∩ A) ⊆ (A × B) ∩ (B × A) ...... (2) 

Now by the definition of set we can say that, from (1) and (2), 

(A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A) [Proved]



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