Complement

Let A be an event in the sample space, S. The set of basic outcomes of a random experiment belonging to S but not to A is called the complement of A and is denoted by

Example: A die is rolled. Let A be the event “Number resulting is even” and B the even “Number resulting is at least 4.” Then

                   A = {2, 4, 6}  and  B = {4, 5, 6}.
Find the complement of each event, the intersection and the union of A and B, and the intersection of  and B.
Solution: The complements of these events are, respectively

The intersection of A and B is the event “Number resulting is either even or at least 4, or both” and so

The union of A and B is the event “Number resulting is either even or at least 4, or both” and so

Note also that the events A and are mutually exclusive, since their intersection is the empty set, and collectively exhaustive, because their union is the sample space S; that is,

Example: Roll of a single die (Unions, Intersections, and Complements)
A die is rolled. Let A be the event “Number resulting is even” and B the event “Number resulting is at least 4.” Then
                    A = {2, 4, 6}       and B = {4, 5, 6}

Fine the complement of each event, the intersection and the union of A and B, and the intersection of   and B.
Solution: The complements of these events are, respectively,

                = {1, 3, 5}   and   = {1, 2, 3}

The intersection of A and B is the event “Number resulting is both even and at least 4” and so

The union of A and B is the event “Number resulting is either even or at least 4, or both” and so

Note also that the events A and  are mutually exclusive, since their intersection is the empty set, and collectively exhaustive, because their union is the sample space S; that is,

 Example: Dow-Jones Industrial Average (Unions, Intersections and Complements).
We will designate four basic outcomes for the Dow-Jones Industrial average over 2 consecutive days:
             Q1 : Dow-Jones average rises on both days.
             Q2 : Dow-Jones average rises on the first day but does not rise on the
                     second day.
             Q3 : Dow-Jones average does not rise on the first day but rises on the
                     second day.
             Q4 : Dow-Jones average does not rise on either day.

Clearly, one of these outcomes must occur, but not more than one can occur at the same time. We can therefore write the sample space as S = {Q1, Q2, Q3, Q4}.

Now, we will consider these two events:

            A : Dow-Jones average rises on the first day.
            B : Dow-Jones average rises on the second day.

Find the intersection, union, and complement of A and B.

Solution: We see that A occurs if either Q1 or Q2 occurs, and, thus,

                 A = {Q1, Q2}           and        B = {Q1, Q3}
The intersection of A and B is the event “Dow-Jones average rises on the first day and rises on the second day.” This is the set of all outcomes belonging to both A and B. Thus,

Finally, the complement of A is the event “Dow-Jones average does not rise on the first day.” This is the set of all basic outcomes in the sample space, S, that do not belong to A. Hence,