Page 25 - Fundamentals of Probability and Statistics for Engineers

8 Fundamentals of Probability and Statistics for Engineers

2.1 ELEMENTS OF SET THEORY

Our interest in the study of a random phenomenon is in the statements we can
make concerning the events that can occur. Events and combinations of events
thus play a central role in probability theory. The mathematics of events is
closely tied to the theory of sets, and we give in this section some of its basic
concepts and algebraic operations.
A set is a collection of objects possessing some common properties. These
objects are called elements of the set and they can be of any kind with any
specified properties. We may consider, for example, a set of numbers, a set of
mathematical functions, a set of persons, or a set of a mixture of things. Capital
, , , , , . . . shall be used to denote sets, and lower-case letters
letters A B C
, , , c !
a b , ,...to denote their elements. A set is thus described by its elements.
Notationally, we can write, for example,
A ˆf1; 2; 3; 4; 5; 6g;

which means that set A has as its elements integers 1 through 6. If set B contains
two elements, success and failure, it can be described by

B ˆfs; f g;

f
s
where and are chosen to represent success and failure, respectively. For a set
consisting of all nonnegative real numbers, a convenient description is
C ˆfx : x 0g:
We shall use the convention
a 2 A 2:1†

to mean ‘element a belongs to set A ’.
A set containing no elements is called an empty or null set and is denoted by . ;
We distinguish between sets containing a finite number of elements and those
having an infinite number. They are called, respectively, finite sets and infinite
sets. An infinite set is called enumerable or countable if all of its elements can be
arranged in such a way that there is a one-to-one correspondence between them
and all positive integers; thus, a set containing all positive integers 1, 2, .. . is a
simple example of an enumerable set. A nonenumerable or uncountable set is one
where the above-mentioned one-to-one correspondence cannot be established. A
simple example of a nonenumerable set is the set C described above.
If every element of a set A is also an element of a set B, the set A is called
a subset of B and this is represented symbolically by

A B or B A: 2:2†

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