6 SETS Chapter 1
and are not, in the set. On the other hand, if the set to be described is
large, the roster method can be impractical or impossible to employ. Clearly
we would want to describe the set F = {xlx is a natural number and
x 2 lo8) by the rule method, although it's theoretically possible to list all
the elements, whereas the set I = (x (x is a real number and 0 2 x 5 1)
cannot even theoretically be described by the roster method. Until we give
a rigorous, or mathematically correct, definition in Article 8.3, we will view
an infinite set as one that cannot, even theoretically, be described by the
roster method. Stated differently, the elements of an infinite set are impos-
sible to exhaust, and so cannot be listed. A finite set, on the other hand, is one
that is not infinite. The set F, defined earlier, is finite, whereas I is infinite
since it has the property that, between any two distinct elements of I, there
is another element of I. A set may fail to have this property and still be
infinite; the set of all positive integers is infinite because, whenever n is a
positive integer, so is n + 1.
A widely used hybrid of the roster and rule methods is employed to de-
scribe both finite and infinite sets. The notation Q = (1, 3, 5,... ,97, 99)
or T = (10, 20, 30, 40,.. .) implicitly uses the rule method by establish-
ing a pattern in which the elements occur. It uses the appearance of the
roster method, with the symbol "... " being read "and so on" in the case
of an infinite set such as T and "... , " meaning "and so on, until" for a
finite set like Q. As with any application of the rule method, there is a
danger of misinterpretation if too little or unclear information is given. As
one example, the notation (1, 2,.. .) may refer to the set (1, 2, 3,4, 5,.. .)
of all positive integers or to the set (1, 2, 4, 8, 16,.. .) of all nonnegative
powers of 2. On the other hand, given the earlier pattern descriptions of
Q and T, most readers would agree that 47 E Q, 2 & Q, 50 E T, 50'' E T, and
15 # T.
There is one other important connection between the roster method and
the rule method. In a number of mathematical situations solving a problem
means essentially to convert a description of a set by the rule method into
a roster method description. In this context we often refer to the roster
representation as the solution set of the original problem (see Exercise 1).UNIVERSAL SETS
Although the idea of a "universal set" in an absolute sense, that is, a set
containing all objects, leads to serious logical difficulties (explored in Ex-
ercise 10) and so is not used in set theory, the concept, when applied in a
more limited sense, has considerable value. For our purposes a universal
set is the set of all objects under discussion in a particular setting.
A universal set will often be specified at the start of a problem in-
volving sets (in this text the letter U will be reserved for this purpose),
whereas in other situations a universal set is more or less clearly, but
implicitly, understood as background to a problem. We did the latter when