Show that B ∪ (∩i∈IAi) = ∩i∈I(B ∪ Ai)

Proof:

⊆: Suppose x ∈ B ∪ (∩i∈IAi)
   We will show that x ∈ ∩i∈I(B ∪ Ai)

   From x ∈ B ∪ (∩i∈IAi), we get x ∈ B ∨ ∀i∈I(x ∈ Ai)
   Which means ∀i∈I(x ∈ B ∨ x ∈ Ai)
   Thus x ∈ ∩i∈I(B ∪ Ai)

⊇: Suppose x ∈ ∩i∈I(B ∪ Ai)
   We will show that x ∈ B ∪ (∩i∈IAi)

   From x ∈ ∩i∈I(B ∪ Ai), we get ∀i∈I(x ∈ B ∨ x ∈ Ai)
   Which means x ∈ B ∨ ∀i∈I(x ∈ Ai)
   Thus x ∈ B ∪ (∩i∈IAi)

# posted by rot13(Unafba Pune) @ 10:27 AM