A Probability Course for the Actuaries A Preparation for Exam P/1

20 BASIC OPERATIONS ON SETS
where the ’or’ is inclusive.(See Figure 2.2(a))
Figure 2.2
The above definition can be extended to more than two sets. More precisely,
if A 1 , A 2 , · · · , are sets then
∪ ∞ n=1A n = {x|x ∈ A i for some i ∈ N}.
The intersection of A and B is the set (See Figure 2.2(b))
A ∩ B = {x|x ∈ A and x ∈ B}.
Example 2.3
Express each of the following events in terms of the events A, B, and C as
well as the operations of complementation, union and intersection:
(a) at least one of the events A, B, C occurs;
(b) at most one of the events A, B, C occurs;
(c) none of the events A, B, C occurs;
(d) all three events A, B, C occur;
(e) exactly one of the events A, B, C occurs;
(f) events A and B occur, but not C;
(g) either event A occurs or, if not, then B also does not occur.
In each case draw the corresponding Venn diagram.
Solution.
(a) A ∪ B ∪ C
(b) (A ∩ B c ∩ C c ) ∪ (A c ∩ B ∩ C c ) ∪ (A c ∩ B c ∩ C) ∪ (A c ∩ B c ∩ C c )
(c) (A ∪ B ∪ C) c = A c ∩ B c ∩ C c
(d) A ∩ B ∩ C
(e) (A ∩ B c ∩ C c ) ∪ (A c ∩ B ∩ C c ) ∪ (A c ∩ B c ∩ C)
(f) A ∩ B ∩ C c