0
Prerequisites
All Chapter topics listed in this chapter are covered in A Primer of Abstract Mathematics by
Robert B. Ash, MAA 1998.
0.1
Elementary Number Theory
The greatest common divisor of two integers can be found by the Euclidean algorithm,
which is reviewed in the exercises in Section 2.5. Among the important consequences of
the algorithm are the following three results.
0.1.1
If d is the greatest common divisor of a and b, then there are integers s and t such that
sa + tb = d. In particular, if a and b are relatively prime, there are integers s and t such
that sa + tb = 1.
0.1.2
If a prime p divides a product a1 an of integers, then p divides at least one ai
0.1.3
Unique Factorization Theorem
If a is an integer, not 0 or 1, then
(1) a can be written as a product p1 pn of primes.
(2) If a = p1 pn = q1 qm , where the pi and qj are prime, then n = m and, after
renumbering, pi = qi for all i.
[We allow negative primes, so that, for example, 17 is prime. This is consistent with the
general denition of prime element in an integral domain; see Section 2.6.]
1
2
0.1.4
CHAPTER 0. PREREQUISITES
The Integers Modulo m
If a and b are integers and m is a positive integer 2, we write a b mod m, and say
that a is congruent to b modulo m, if a b is divisible by m. Congruence modulo m
is an equivalence relation, and the resulting equivalence classes are called residue classes
mod m. Residue classes can be added, subtracted and multiplied consistently by choosing
a representative from each class, performing the appropriate operation, and calculating
the residue class of the result. The collection Zm of residue classes mod m forms a
commutative ring under addition and multiplication. Zm is a eld if and only if m is
prime. (The general denitions of ring, integral domain and eld are given in Section 2.1.)
0.1.5
(1) The integer a is relatively prime to m if and only if a is a unit mod m, that is, a has
a multiplicative inverse mod m.
(2) If c divides ab and a and c are relatively prime, then c divides b.
(3) If a and b are relatively prime to m, then ab is relatively prime to m.
(4) If ax ay mod m and a is relatively prime to m, then x y mod m.
(5) If d = gcd(a, b), the greatest common divisor of a and b, then a/d and b/d are relatively
prime.
(6) If ax ay mod m and d = gcd(a, m), then x y mod m/d.
(7) If ai divides b for i = 1, . . . , r, and ai and aj are relatively prime whenever i = j , then
the product a1 ar divides b.
(8) The product of two integers is their greatest common divisor times their least common
multiple.
0.1.6
Chinese Remainder Theorem
If m1 , . . . , mr are relatively prime in pairs, then the system of simultaneous equations
x bj mod mj , j = 1, . . . , r, has a solution for arbitrary integers bj . The set of solutions
forms a single residue class mod m=m1 mr , so that there is a unique solution mod m.
This result can be derived from the abstract form of the Chinese remainder theorem;
see Section 2.3.
0.1.7
Eulers Theorem
The Euler phi function is dened by (n) = the number of integers in {1, . . . , n} that
are relatively prime to n. For an explicit formula for (n), see Section 1.1, Problem 13.
Eulers theorem states that if n 2 and a is relatively prime to n, then a(n) 1 mod n.
0.1.8
Fermats Little Theorem
If a is any integer and p is a prime not dividing a, then ap1 1 mod p. Thus for any
integer a and prime p, whether or not p divides a, we have ap a mod p.
For proofs of (0.1.7) and (0.1.8), see (1.3.4).
0.2. SET THEORY
0.2
3
Set Theory
0.2.1
A partial ordering on a set S is a relation on S that is reexive (x x for all x S ),
antisymmetric (x y and y x implies x = y ), and transitive (x y and y z implies
x z ). If for all x, y S , either x y or y x, the ordering is total.
0.2.2
A well-ordering on S is a partial ordering such that every nonempty subset A of S has a
smallest element a. (Thus a b for every b A).
0.2.3
Well-Ordering Principle
Every set can be well-ordered.
0.2.4
Maximum Principle
If T is any chain (totally ordered subset) of a partially ordered set S , then T is contained
in a maximal chain M . (Maximal means that M is not properly contained in a larger
chain.)
0.2.5
Zorns Lemma
If S is a nonempty partially ordered set such that every chain of has S an upper bound
in S , then S has a maximal element.
(The element x is an upper bound of the set A if a x for every a A. Note that
x need not belong to A, but in the statement of Zorns lemma, we require that if A is a
chain of S , then A has an upper bound that actually belongs to S .)
0.2.6
Axiom of Choice
Given any family of nonempty sets Si , i I , we can choose an element of each Si .
Formally, there is a function f whose domain is I such that f (i) Si for all i I .
The well-ordering principle, the maximum principle, Zorns lemma, and the axiom of
choice are equivalent in the sense that if any one of these statements is added to the basic
axioms of set theory, all the others can be proved. The statements themselves cannot be
proved from the basic axioms. Constructivist mathematics rejects the axiom of choice
and its equivalents. In this philosophy, an assertion that we can choose an element from
each Si must be accompanied by an explicit algorithm. The idea is appealing, but its
acceptance results in large areas of interesting and useful mathematics being tossed onto
the scrap heap. So at present, the mathematical mainstream embraces the axiom of
choice, Zorns lemma et al.
4
0.2.7
CHAPTER 0. PREREQUISITES
Proof by Transnite Induction
To prove that statement Pi holds for all i in the well-ordered set I , we do the following:
1. Prove the basis step P0 , where 0 is the smallest element of I .
2. If i > 0 and we assume that Pj holds for all j < i (the transnite induction
hypothesis), prove Pi .
It follows that Pi is true for all i.
0.2.8
We say that the size of the set A is less than or equal to the size of B (notation A s B ) if
there is an injective map from A to B . We say that A and B have the same size (A =s B )
if there is a bijection between A and B .
0.2.9
Schrder-Bernstein Theorem
o
If A s B and B s A, then A =s B . (This can be proved without the axiom of choice.)
0.2.10
Using (0.2.9), one can show that if sets of the same size are called equivalent, then s
on equivalence classes is a partial ordering. It follows with the aid of Zorns lemma that
the ordering is total. The equivalence class of a set A, written |A|, is called the cardinal
number or cardinality of A. In practice, we usually identify |A| with any convenient
member of the equivalence class, such as A itself.
0.2.11
For any set A, we can always produce a set of greater cardinality, namely the power set 2A ,
that is, the collection of all subsets of A.
0.2.12
Dene addition and multiplication of cardinal numbers by |A| + |B | = |A B | and |A||B | =
|A B |. In dening addition, we assume that A and B are disjoint. (They can always be
disjointized by replacing a A by (a, 0) and b B by (b, 1).)
0.2.13
If 0 is the cardinal number of a countably innite set, then 0 + 0 = 0 0 = 0 . More
generally,
(a) If and are cardinals, with and innite, then + = .
(b) If = 0 (i.e., is nonempty), and is innite, then = .
0.2.14
If A is an innite set, then A and the set of all nite subsets of A have the same cardinality.
0.3. LINEAR ALGEBRA
0.3
5
Linear Algebra
It is not feasible to list all results presented in an undergraduate course in linear algebra.
Instead, here is a list of topics that are covered in a typical course.
1. Sums, products, transposes, inverses of matrices; symmetric matrices.
2. Elementary row and column operations; reduction to echelon form.
3. Determinants: evaluation by Laplace expansion and Cramers rule.
4. Vector spaces over a eld; subspaces, linear independence and bases.
5. Rank of a matrix; homogeneous and nonhomogeneous linear equations.
6. Null space and range of a matrix; the dimension theorem.
7. Linear transformations and their representation by matrices.
8. Coordinates and matrices under change of basis.
9. Inner product spaces and the projection theorem.
10. Eigenvalues and eigenvectors; diagonalization of matrices with distinct eigenvalues,
symmetric and Hermitian matrices.
11. Quadratic forms.
A more advanced course might cover the following topics:
12. Generalized eigenvectors and the Jordan canonical form.
13. The minimal and characteristic polynomials of a matrix; Cayley-Hamilton theorem.
14. The adjoint of a linear operator.
15. Projection operators.
16. Normal operators and the spectral theorem.