24 SETS Chapter 1
Figure 1.2 Graphic interpretation of the mean value theorem:
f '
to the conjecture that this relationship holds for any set B. In trigonometry
the specific facts that sin (2n) = 0, sin (44 = 0, and sin (- 3n) = 0 suggest
a possible general theorem, namely, sin (kn) = 0 for any integer k, which,
upon being proved, encompasses an infinite number of particular cases. In
elementary calculus the facts that d/dx(x2) = 2x, d/dx(x3) = 3x2, and
d/dx(x4) = 4x3, again suggest the possible formulation of a rule that includes
these formulas and yields the answer for infinitely many similar differenti-
ation problems.
Another type of evidence from which mathematical conjectures are
frequently drawn is a picture. For example, the mean value theorem of
calculus has a lengthy and complicated statement when expressed in words
only, but becomes simple in concept when that statement is accompanied
by a picture, as shown in Figure 1.2. In addition, many of the applied prob-
lems of calculus, such as "max-min" and "related rate" problems, have as
a standard part of their approach the step "draw a picture of the physical
situation described in the problem."
The kind of picture most frequently used to seek out theorems, or to test
conjectures about possible theorems, in set theory is the Venn diagram.
VENN DIAGRAMS
Sets pictured by Venn diagrams appear as labeled circles, inside a rectangle
that represents the universal set U. Most often, such diagrams will involve
one, two or three circles, with various markings used to match regions in
the diagram with sets formed, by employing the operations described earlier,
from the sets represented by the circles. As one example, given two sets A
and B, the set A n B is represented by the shaded region of the diagram in
Figure 1.3.