Robert Bartoszynski - Probability and Statistical Inference

An alternative terminology here is that implies (or entails ) .

Definition 1.3.2Two events and are said to be equal , , if and .

It follows that two events are equal if they consist of exactly the same sample points.

Consider two tosses of a coin, and the corresponding sample space consisting of four outcomes: HH, HT, TH, and TT. The event “heads in the first toss” HH, HT} is contained in the event “at least one head” HH, HT, TH}. The events “the results alternate” and “at least one head and one tail” imply one another, and hence are equal.

Definition 1.3.3The set containing no elements is called the empty set and is denoted by . The event corresponding to is called a null ( impossible ) event.

The reader may wonder whether it is correct to use the definite article in the definition above and speak of “ the empty set,” since it would appear that there may be many different empty sets. For instance, the set of all kings of the United States and the set of all real numbers such that are both empty, but one consists of people and the other of numbers, so they cannot be equal. This is not so, however, as is shown by the following formal argument (to appreciate this argument, one needs some training in logic). Suppose that and are two empty sets. To prove that they are equal, one needs to prove that and . Formally, the first inclusion is the implication: “if belongs to , then belongs to .” This implication is true, because its premise is false: there is no that belongs to . The same holds for the second implication, so .

We now give the definitions of three principal operations on events: complementation , union , and intersection .

Definition 1.3.4The set that contains all sample points that are not in the event will be called the complement of and denoted , to be read also as “not .”

Definition 1.3.5The set that contains all sample points belonging either to or to (so possibly to both of them) is called the union of and and denoted , to be read as “ or .”

Definition 1.3.6The set that contains all sample points belonging to both and is called the intersection of and and denoted .