Question :
Show that if the event A is completely independent of events B and C, then A is independent of logical sum
I guess "completely independent" is what is also called "mutually independent" (by opposition to "pairwise independent"), i.e. it means that (same with A,C and B,C), and .
Then the property is proved as follows (for instance):
, and the three (main) events on the right-hand side are disjoints, hence
,
and you can compute the latter probabilities using (and similarly for the last term) and the assumptions (recalled above).
You should get , or in fact , which is the same...