For the sake of clarity, we gather together in these preliminary pages some of the more
important mathematical and statistical concepts with which we assume prior familiarity, concentrating
especially on those for which notational conventions unfortunately vary from one
textbook to another.
Mathematical and technical terms appear in boldface where they are first introduced or
defined; the corresponding page number in the index is also in boldface.
Economics students who are new to formal mathematics should be aware of common pitfalls
of flawed logic, in particular with the importance of presenting the parts of a definition
in the correct order and with the process of proving a theorem by arguing from the assumptions
to the conclusions. Familiarity with various approaches to proofs is assumed, though
the principles of proof by contradiction, proof by contrapositive and proof by induction are
described when these methods are first used.1 Similarly, mathematics students, who may
be familiar with many of the mathematics topics covered, should think about the nature,
subject matter and scientific methodology of economics before starting to work through the
book.
Readers should be familiar with the expressions “such that” and “subject to” (both often
abbreviated “s.t.”) and “if and only if” (abbreviated as “iff”), and also with their meanings
and use. The symbol ∀ is mathematical shorthand for “for all” and ∃ is mathematical shorthand
for “there exists”. The expression iff signifies a necessary and sufficient condition, or
equivalence. Briefly, Q is necessary for P if P implies Q; and similarly P is sufficient for
Q if P implies Q. Furthermore, P implies Q if and only if the contrapositive, “not Q”
implies “not P”, is true. We shall sometimes use an alternative symbol for a necessary and
sufficient condition, namely ⇔, which signifies that the truth of the left-hand side implies
the truth of the right-hand side and vice versa, and also that the falsity of the left-hand side
implies the falsity of the right-hand side and vice versa. Other logical symbols used are ⇒,
which means “implies”, and⇐, which means “is implied by” or “follows from”.