Page 28 - Probability and Statistical Inference

1. Nations of Probability 5

Example 1.2.2 (Example 1.2.1 Continued) One can verify that A∪B = {1,
5, 7, 11, 13}, B∪C = S, but C and D are mutually exclusive. Also, A and D are
mutually exclusive. Note that A = {3, 9, 11, 13}, Α∆C = {1, 3, 9} and A∆B =
c
{11, 13}. !
Now consider {A ; i ∈ I}, a collection of subsets of S. This collection may
i
be finite, countably infinite or uncountably infinite. We define

The equation (1.2.3) lays down the set operations involving the union and
intersection among arbitrary number of sets. When we specialize I = {1, 2, 3,
...} in the definition given by (1.2.3), we can combine the notions of countably
infinite number of unions and intersections to come up with some interesting
sets. Let us denote

Interpretation of the set B: Here the set B is the intersection of the collec-
tion of sets , j = 1. In other words, an element x will belong to B if and
only if x belongs to A for each j = 1 which is equivalent to saying that
i
there exists a sequence of positive integers i < i < ... < i < ... such that x ∈
k
1
2
A for all k = 1, 2, ... . That is, the set B corresponds to the elements which
ik
are hit infinitely often and hence B is referred to as the limit (as n → ∞)
supremum of the sequence of sets A , n = 1, 2, ... .
n
Interpretation of the set C: On the other hand, the set C is the union of the
collection of sets A j = 1. In other words, an element x will belong to C if
i
and only if x belongs to A for some j ≥ 1 which is equivalent to saying that
i
x belongs to A , A , ... for some j ≥ 1. That is, the set C corresponds to the
j
j+1
elements which are hit eventually and hence C is referred to as the limit
(as n → ∞) infimum of the sequence of sets A , n = 1, 2, ... .
n
Theorem 1.2.1 (DeMorgan’s Law) Consider {A ; i ∈ I}, a collection of
subsets of S. Then, i

Proof Suppose that an element x belongs to the lhs of (1.2.5). That is, x ∈ S
but x ∉ ∪ A , which implies that x can not belong to any of the
i∈I i