Bridge to Abstract Mathematics: Mathematical Proof and Structures

1.2 OPERATIONS ON SETS 23

with reference to two perpendicular lines, the x axis and the y axis. I x J

is the rectangle, open above and below, closed left and right, as illustrated

in Figure l.lb.

For other examples, suppose that L = (xlx is a student in Math 197)
and M = (x 1 x is a possible final grade in a course} = (A, B, C, D, E}.
Can you describe generally what the set L x M looks like? In another vein,
if A = (1,2,3}, can you describe A x a? 0 x A? What general results are
suggested by this example? (See Fact 7 (49), Article 1.4.)
Like other operations on sets, cartesian products can be combined with
other set operations to form expressions like (A u B) x C. See Exercise 4
for other examples.
The cartesian product will be of particular importance in Chapters 7 and
8 where we study relations on sets, including equivalence relations and
functions. It should be noted that a rigorous set theoretic definition of an
ordered pair (omitted from the preceding informal discussion) is possible
and may be found in Exercise 10, Article 4.1.
Finally, note that the use of (,) as notation for ordered pairs, while also
denoting open intervals, is an example of the ambiguity in mathematical
notation alluded to earlier.

TOWARD MATHEMATICAL GENERALIZATION

Throughout this article and Article 1.1 you have been encouraged to specu-
late on possible general theorems of set theory as suggested by examples.
For instance, you may have already conjectured that the union of two sets
A and B is a superset of both A and B (between Examples 1 and 2), that the
intersection of a set with its complement is the empty set (following Example
4), or that the intersection of two intervals is always an interval (following
Example 2).
The mode of thinking we're trying to foster through our questions is the
first half of a two-part process that is really the essence of mathematics! By
looking at particular situations, the mathematician hopes to be able to for-
mulate general conjectures that, if true, settle the question at hand about all
other particular cases. The first step into the world of the mathematician
is to form the intellectual habit of looking beyond the answer to the exam-
ple at hand to possible general reasons for that answer. We will continue
to discuss this first step, the formulation of conjectures, in Article 1.3 and
will begin to pursue the second step, the construction of proofs that turn
our conjectures into theorems in Chapter 4.
But for now we consider further the types of evidence on which mathe-
matical conjectures might be based. We have already seen one type of such
evidence, namely, computational solutions to particular problems. As fur-
ther examples of this approach, the fact that B n B = B, which was dis-
cussed earlier for the particular set B = (1,4,7, 10, 13, 161, should lead