1.1 BASIC DEFINITIONS AND NOTATION 5Two important facts are: (1) the order in which elements are listed is
irrelevant and (2) an object should be listed only once in the roster, since
listing it more than once does not change the set. As an example, the set
{ 1, 1,2) is the same as the set {1,2) (so that the representation (1, 1,2)
is never used) which, in turn, is the same as the set (2, 1).The rule, or description, method. We describe a set in terms of one or more
properties to be satisfied by objects in the set, and by those objects only.
Such a description is formulated in so-called set-builder notation, that is,
in the form A = {x 1 x satisfies some property or properties), which we read
"A is the set of all objects x such that x satisfies... ." Typical representa-
tions of sets by the rule method are:
C = {x 1 x is a natural number and x 5 100.)
or D = {x 1 x is the name of a state in the United States beginning
with the letter M.)
or X = {x lx is a male citizen of the United States.)
In all these examples the vertical line is read "such that" and the set is un-
derstood to consist of & objects satisfying the preceding description, and
only those objects. Thus 57 E C, whereas 126 4 C. The set D can also be
described by the roster method, namely, as the set {Maine, Maryland,
Massachusetts, Michigan, Minnesota, Missouri, Mississippi, Montana).
Although it's true that Maine E D, it would be false to say that D = {Maine);
that is, the description of D must not be misinterpreted to mean that D has
only one element. The same is true of the set X which is a very large set,
difficult to describe by the roster method.
It is in connection with the description method that "well definedness"
comes into play. The rule or rules used in describing a set must be (1) mean-
ingful, that is, use words and/or symbols with an understood meaning
and (2) specific and definitive, as opposed to vague and indefinite. Thus
descriptions like G = {x(x is a goople) or E = {xlx!* & 3) or Z = {xlx is
a large state in the United States) do not define sets. The descriptions of. G and E involve nonsense symbols or words, while the description of Z
gives a purely subjective criterion for membership. On the other hand, a set
may be well defined even though its membership is difficult to determine or
not immediately evident from its description (see Exercise 3).
COMPARISON OF THE ROSTER AND RULE METHODS.
FINITE AND INFINITE SETS
The roster method has the obvious advantage of avoiding the problem of
deciding well definedness. Whenever it's used (provided the objects named
as elements have meaning), there can be no doubt as to which objects are,