For the inclusion-exclusion identity of Miscellanea 1.8.1:
(a) Derive both Boole's and Bonferroni's Inequality from the inclusion-exclusion identity.
(b) Show that the Pi satisfy Pi ≥ Pj if i > j and that the sequence of bounds in Miscellanea 1.8.1 improves as the number of terms increases.
(c) Typically as the number of terms in the bound increases, the bound becomes more useful. However, Schwager (1984) cautions that there are some cases where there is not much improvement, in particular if the AiS are highly correlated. Examine what happens to the sequence of bounds in the extreme case when Ai = A for every i. (See Worsley 1982 and the correspondence of Worsley 1985 and Schwager 1985.)