Question

Let A and B be two sets. show that the set A*B and B*A have an element in common iff the sets A and B have an element in common .​

Answer 1

Step-by-step explanation:

thanks for your question

Answer 2

Important formulas :

A ∪ B = B ∪ A

A ∪ (B ∪ C) = (A ∪ B) ∪ C.

Let A and B have x elements in common .

n { (A ∩ B) × (B ∪ A) } = n { (A × B) ∩ (B × A)

⇒ n { (A ∩ B) × (B ∪ A) } = n { (A ∩ B) } × n{ (A ∩ B) }

⇒ n { (A ∩ B) × (B ∪ A) } = k × k ⇒ k²

Hence A × B and B × A have an element in common . The number of elements is k² provided that k is the number of elements in common in A and B .