calculus book - Mathematics and Computer Science

X Y
X ∪ Y
1.3. LOGIC 9
X Y
X ∩ Y
X Y
X \ Y
Figure 1.2: Venn diagrams for simple set operations.
In this book, an infinite index set is usually the set of positive integers,
as in
∞
X = (− 1
n=1
n
, 1
n ).
A real number x is an element of X exactly when −1/n < x < 1/n for
every positive integer n. We will see that x = 0 is the only real number
with this property, so X = {0}. Additional examples of infinite set
operations are given in Exercise 2.10.
1.3 Logic
As mentioned, mathematicians do not really find “objective truths”, but
instead derive logical conclusions by starting with assumptions called
hypotheses. In this section we begin the study of logical deduction,
emphasizing the linguistic differences with ordinary English.
Mathematicians use generally agreed-upon axioms (foundational assumptions)
and rules of logical deduction. In this book, the axioms are
the Zermelo-Frankel-Choice axioms of set theory, and the rules of deduction
are those of Aristotelean logic.
Throughout this section, a running sequence of examples is presented
to illustrate the concepts being introduced. Mathematics aims
for generality, but the human mind often favors particulars, and it is
these that make mathematics directly useful. The aim of mathematics
is precise thinking, not generality for its own sake. That said, abstraction
(which fosters generality) has a definite purpose: To extract the
essential features of a problem and ignore extraneous details. Precision
is important because intuition (especially regarding the infinite)
is often misleading, sometimes blatantly wrong. Logical deduction is