27. A point is chosen at random from the interval (−1, 1). Let E1 be the event that it falls in the interval (−1/2, 1/2), E2 be the event that it falls in the interval (−1/4, 1/4) and, in general, for 1 ≤ i < ∞, Ei be the event that the point is in the interval (−1/2i, 1/2i).

Find S

∞ i

=1 Ei and T

∞ i

=1 Ei.

28. Prove De Morgan’s second law, (AB)c = Ac ∪ Bc,

(a) by elementwise proof;

(b) by applying De Morgan’s first law to Ac and Bc.
29. Let A and B be two events. Prove the following relations by the elementwise method.

(a) (A − AB) ∪ B = A ∪ B.

(b) (A ∪ B) − AB = ABc ∪ AcB.
30. Let {An}∞n=1 be a sequence of events. Prove that for every event B,

(a) B ????S ∞ i =1 Ai

= S ∞ i =1 BAi.

(b) B S????T ∞ i =1 Ai

= T ∞ i =1(
B ∪
Ai)
.