Given ∀i∈IAi ⊆ X, show that (∪i∈IAi)c = ∩i∈IAic

Proof:

⊆: Suppose x ∈ (∪i∈IAi)c
   We will show that x ∈ ∩i∈IAic

   From x ∈ (∪i∈IAi)c, we get ¬∃i∈I(x ∈ Ai)
   It follows that ∀i∈I(x ∉ Ai), i.e., ∀i∈I(x ∈ Aic)
   Thus x ∈ ∩i∈IAic

⊇: Suppose x ∈ ∩i∈IAic
   We will show that x ∈ (∪i∈IAi)c

   From x ∈ ∩i∈IAic, we get ∀i∈I(x ∈ Aic), i.e., ∀i∈I(x ∉ Ai)
   It follows that ¬∃i∈I(x ∈ Ai)
   Thus x ∈ (∪i∈IAi)c

# posted by rot13(Unafba Pune) @ 10:39 AM