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) @ 10:04 AM