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Algebra of Events

Algebra of Events

Definition 2.10. (Union of two events.) A complex event which consists of the
occurrence of at least one of the events A and B is called the union or the sum
of two events A and B. The union of the events A and B is denoted A ∪ B. For
exclusive events A and B we also use the notation A + B.
The union of several event A 1 , A 2 , . . . , A n is defined in the same way, and is
∪
denoted by A k , ✓
k
Definition 2.11. (Intersection of two events.) The joint appearance of two events
A and B is called the intersection or the product of these two events and is
denoted A ∩ B or AB.
n
∩
By the intersection of several events A 1 , A 2 , . . . , A n denoted by A k (k =
k=1
1, 2, . . . , n) we mean the event consisting of the occurrence of all events. ✓

Given two events A and B, by the difference A − B we mean the event in which A occurs but
not B.
Properties of unions and intersections. The operations of unions and intersections of
events posses some properties which are similar to those of addition and multiplication of
numbers.

1. The union and intersection of events are commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A.

2. The union and intersection of events are associative:

A ∪ B) ∪ C = A ∪ (B ∪ C) = (A ∪ C) ∪ B = A ∪ B ∪ C,

(AB)C = A(BC) = (AC)B = ABC.
3. The union and intersection of events are distributive:

(A ∪ B)C = AC ∪ BC.

All these properties follow directly from the definition of union and intersection. Thus, (A ∪
B)C means the joint occurrence of the event C with the event A, or with the event B, or with A
and B together. The event AC ∪ BC also means the occurrence of either C together with A, or C
together with B, or C together with AB.
But not all laws of addition and multiplication of numbers are valid for union and intersection
of events. Thus, for instance, the events A∪A and AA and AA evidently coincide with A. Therefore
A ∪ A = AA = A.
Complementary events. Non-occurrence of A, which is denoted by A, is the event
complementary to the event A.
¯
It is easy to see that the event A is complementary to the event A : A = A.
Gain and loss in a game, failure of a device in a given time interval and its faultless functioning
in the same time interval, are examples of complementary events.
It is evident that complementary events are exclusive, and their union is a certain event:
¯
¯
AA = ∅, A ∪ A = S,
where S is the certain event and ∅ is the impossible event.
It is also clear that
A ∪ ∅ = A, A∅ = ∅, A ∪ S = S, AS = A.

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