a) Prove that the following De Morgan's laws hold for any sets A, B and C: A\(B?C) = (A\B)?(A\C), A\(B?C) = (A\B)?(A\C).

b) Show that intersection of infinitely many open sets is not necessarily open. Use De Morgan's law to prove that union of in?nitely many closed sets is not necessarily closed.

Problem 2 . 2 ) Prove that the following De Morgan's laws hold for any sets A, Band C : A\\ ( BUC ) = ( A / B ) n ( AIC ), Al ( BN(" ) = ( A / B ) U (AIC ). b) Recall the example from the lecture , showing that intersection of infinitely many open sets is not necessarily open . Use this example and De Morgan's laws to prove that union of infinitely many closed sets is not necessarily closed