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

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12 BASIC OPERATIONS ON SETS<br />

1 Basic Definitions<br />

We define a set A as a collection of well-defined objects (called elements or<br />

members of A) such that <strong>for</strong> any given object x ei<strong>the</strong>r one (but not both)<br />

of <strong>the</strong> following holds:<br />

• x belongs to A and we write x ∈ A.<br />

• x does not belong to A, and in this case we write x ∉ A.<br />

<strong>Exam</strong>ple 1.1<br />

Which of <strong>the</strong> following is a well-defined set.<br />

(a) The collection of good books.<br />

(b) The collection of left-handed individuals in Russellville.<br />

Solution.<br />

(a) The collection of good books is not a well-defined set since <strong>the</strong> answer to<br />

<strong>the</strong> question “Is My Life a good book” may be subject to dispute.<br />

(b) This collection is a well-defined set since a person is ei<strong>the</strong>r left-handed or<br />

right-handed. Of course, we are ignoring those few who can use both hands<br />

There are two different ways to represent a set. The first one is to list,<br />

without repetition, <strong>the</strong> elements of <strong>the</strong> set. For example, if A is <strong>the</strong> solution<br />

set to <strong>the</strong> equation x 2 − 4 = 0 <strong>the</strong>n A = {−2, 2}. The o<strong>the</strong>r way to represent<br />

a set is to describe a property that characterizes <strong>the</strong> elements of <strong>the</strong> set. This<br />

is known as <strong>the</strong> set-builder representation of a set. For example, <strong>the</strong> set A<br />

above can be written as A = {x|x is an integer satisfying x 2 − 4 = 0}.<br />

We define <strong>the</strong> empty set, denoted by ∅, to be <strong>the</strong> set with no elements. A<br />

set which is not empty is called a nonempty set.<br />

<strong>Exam</strong>ple 1.2<br />

List <strong>the</strong> elements of <strong>the</strong> following sets.<br />

(a) {x|x is a real number such that x 2 = 1}.<br />

(b) {x|x is an integer such that x 2 − 3 = 0}.<br />

Solution.<br />

(a) {−1, 1}.<br />

(b) Since <strong>the</strong> only solutions to <strong>the</strong> given equation are − √ 3 and √ 3 and both<br />

are not integers <strong>the</strong>n <strong>the</strong> set in question is <strong>the</strong> empty set

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