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

12 BASIC OPERATIONS ON SETS
1 Basic Definitions
We define a set A as a collection of well-defined objects (called elements or
members of A) such that for any given object x either one (but not both)
of the following holds:
• x belongs to A and we write x ∈ A.
• x does not belong to A, and in this case we write x ∉ A.
Example 1.1
Which of the following is a well-defined set.
(a) The collection of good books.
(b) The collection of left-handed individuals in Russellville.
Solution.
(a) The collection of good books is not a well-defined set since the answer to
the question “Is My Life a good book” may be subject to dispute.
(b) This collection is a well-defined set since a person is either left-handed or
right-handed. Of course, we are ignoring those few who can use both hands
There are two different ways to represent a set. The first one is to list,
without repetition, the elements of the set. For example, if A is the solution
set to the equation x 2 − 4 = 0 then A = {−2, 2}. The other way to represent
a set is to describe a property that characterizes the elements of the set. This
is known as the set-builder representation of a set. For example, the set A
above can be written as A = {x|x is an integer satisfying x 2 − 4 = 0}.
We define the empty set, denoted by ∅, to be the set with no elements. A
set which is not empty is called a nonempty set.
Example 1.2
List the elements of the following sets.
(a) {x|x is a real number such that x 2 = 1}.
(b) {x|x is an integer such that x 2 − 3 = 0}.
Solution.
(a) {−1, 1}.
(b) Since the only solutions to the given equation are − √ 3 and √ 3 and both
are not integers then the set in question is the empty set