

InterviewSolution
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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