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Chapter 4
Properties of the Integers:
Mathematical Induction
Introduction
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Mathematical induction 數學歸納法 is a technique exhibited
by the subset of positive integers.
Four sets of numbers that are very important in the study
of discrete mathematics and combinatorics are introduced –
namely, the harmonic numbers調和數列, the Fibonacci
numbers費伯納西數列, the Lucas numbers, and the Eulerian
numbers.
When x, y  Z, we know that x + y, xy, x – y  Z. Thus we
say that the set Z is closed under (the binary operations of)
addition, multiplication, and subtraction.
Note that 2, 3  Z but that the rational number 2/3 is not a
member of Z. So the set Z of all integers is not closed
under the binary operation of nonzero division.
A special concentration on primes ( Z+).
4.1 The Well-Ordering Principle:
Mathematical Induction
The Well-Ordering Principle (of positive
integers): Every nonempty subset of Z+
contains a smallest element. (often
saying that Z+ is well ordered)
 The set Z+ is different from the sets Q+
and R+ in that every nonempty subset X
of Z+ contains an integer a such that a  x,
for all x  X.
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Theorem 4.1: Finite Induction Principle,
or Principle of Mathematical Induction.
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Let S(n) denote an open mathematical statement
(or set of such open statements) the involves one
or more occurrences of the variable n, which
represents a positive integer.
If S(1) is true (referred to as the basic step);
and
If whenever S(k) is true (for some particular,
but arbitrarily chosen, k  Z+), then S(k +1)
is true (called the inductive step);
then S(n) is true for all n  Z+.
Mathematical Induction
Proof: see page 164.
 The choice of 1 in the first condition of
Theorem 4.1 is not mandatory. If the true
of S(n0) is for our basic step (a different
starting point!), we can rewrite the Finite
Induction Principle, using quantifiers, as
 [S(n0)  [ k  n0 [S(k)  S(k + 1)]]]
  n  n0 S(n).
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Example 4.1: page 166.
Example 4.3: page 167.
Example 4.4: page 168.
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(This example indicates the effect of the wrong
starting point – n  7 or n  137.)
Example 4.7 (the harmonic numbers
H1, H2, …): page 171.
Example 4.9: page 172.

(Program verification by using mathematical
induction!)
Theorem 4.2: Finite Induction Principle –
Alternative Form.
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Let S(n) denote an open mathematical statement
(or set of such open statements) that involves
one or more occurrences of the variable n, which
represents a positive integer. Also let n0, n1  Z+
with n0  n1.
If S(n0), S(n0 + 1), S(n0 + 2), …, S(n1 – 1),
and S(n1) are true; and
If whether S(n0), S(n0 + 1), …, S(k – 1), and
S(k) are true for some (particular but
arbitrarily chosen) k  Z+, where k  n1,
then the statement S(k + 1) is also true;
then S(n) is true for all n  n0.
Example 4.11: page 175.
Example 4.12: page 175.
4.2 Recursive Definitions
The terms of some integer sequences may have explicit formula
to define, for example,
b0 = 2  0 = 0,
b1 = 2  1 = 2,
b2 = 2  2 = 4,
b3 = 2  3 = 6,
…,
where bn = 2  n for all n  N.
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Other sequences may not have explicit formula, for example,
a0 = 1, a1 = 2, a2 = 3,
and
an = an-1 + a= + an-3,
for all n  N where n  3.
 The way we define for a0, a1, a2, …, an, … is called a recursive
definition. The assignments a0 = 1, a1 = 2, a2 = 3 provide a
base for the recursion. The equation an = an-1 + an-2 + an-3, for all
n  N where n  3, provides the recursive process; it indicates
how to obtain new entries in the sequence from those prior results
we already know.
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Example 4.13 (Generalized Associative
Law for ),
Example 4.14 (Generalized Associative
Law for ),
Example 4.15 (Generalized DeMorgan
Law):
Example 4.16 (the Fibonacci numbers):
see page 183;
the recursive definition of the Fibonacci
numbers can be used to establish many
of the interesting properties that these
numbers exhibit. One of these properties
is as follows:
  n  Z+

n
2
 Fi  Fn  Fn 1
i 0
Proof: see page 184.
Example 4.17 (the Lucas numbers): see
page 184~185.

(One of the interrelations between the Fibonacci
and Lucas numbers is illustrated in the fact that
 n  Z+ .)
Ln  Fn 1  Fn 1
Example 4.18 (the Eulerian numbers):
see page 185~186.
m 1a
 (Note: for a fixed m  Z+, .)
 k  0 m, k  m!
4.3 The Division Algorithm: Prime
Numbers
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Definition 4.1: If a, b  Z and b ≠ 0, we
say that b divides a, and we write b|a, if
there is an integer n such that a = bn.
When this occurs we say that b is a divisor
of a, or a is a multiple of b.
Theorem 4.3: For all a, b, c  Z
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a) 1|a and a|0.
b) [(a|b)  (b|a)]  a = b.
c) [(a|b)  (b|c)]  a|c. d) a|b  a|bx for all x 
Z.
e) If x = y + z, for some x, y, z  Z, and a
divides two of the three
integers x, y, and z, then a divides the remaining
integer.
f) [(a|b)  (a|c)]  a|(bx + cy), for all x, y  Z.
(The expression bx + cy is called a linear
combination of b, c.)
g) For 1in, let ci  Z. If a divides each ci, then
a|(c1x1 + c2x2 + …+ cnxn) , where xi  Z for all
1in.
Example 4.20 and 21: page 190.
Lemma 4.1:
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If n  Z+ and n is composite, then there is a
prime p such p | n.
Theorem 4.4: (Euclid) There are
infinitely many primes.
Theorem 4.5: The Division Algorithm.

If a, b  Z, with b > 0, then there exist
unique q, r  Z with a = qb + r, 0  r < b.
Example 4.22: page 192.
Example 4.24, page 194
Example 4.25
Example 4.26, 4.27: page 196~198.

(Note on one’s and two’s complements!)
4.4 The Greatest Common Divisor:
The Euclidean Algorithm
Definition 4.2:
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For a, b  Z, a positive integer c is said to
be a common divisor of a and b if c|a and
c|b.
Example 4.29: page 200.
Definition 4.3:
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Let a, b  Z, where either a  0, or b  0. Then c 
Z+ is called a greatest common divisor of a, b if
c|a and c|b (that is, c is a common divisor of a, b),
and
for any common divisor d of a and b, we have d|c.
Theorem 4.6:
For all a, b  Z+, there exists a unique c 
Z+ that is the greatest common divisor of
a, b (abbreviated as gcd(a,b)).
 Proof: See page 201.
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From Theorem 4.6 we see that not only
does gcd(a,b) exist but that gcd(a,b) is
also the smallest positive integer we can
write as a linear combination of a and b.
 Integers a and b are called relatively
prime when gcd(a,b) = 1 – that is, when
there exist x, y  Z with ax + by = 1.
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See Example 4.30: page 201.
Theorem 4.7:
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Euclidean Algorithm. If a, b  Z+, we apply the
division algorithm as follows:
Proof: See page 202.
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(The computing procedure is shown in Fig 4.9.)
Example 4.31: page 203.
Example 4.32: page 203.
Theorem 4.8:
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If a, b, c,  Z+, the Diophantine equation
ax + by = c has an integer solution x = x0,
y = y0 if and only if gcd(a,b) divides c.
Definition 4.4:
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For a, b, c,  Z+, c is called a common
multiple of a, b if c is a multiple of both a
and b. Furthermore, c is the least common
multiple of a, b if it is the smallest of all
positive integers that are common
multiples of a, b. We denote c by lcm(a,b).
Example 4.36: page 205.
Theorem 4.9:
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Let a, b, c,  Z+, with c = lcm(a,b). If d is a
common multiple of a and b, then c|d.
Theorem 4.10:
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For all a, b  Z+, ab = lcm(a,b) × gcd(a,b).
Example 4.37: page 206.
4.5 The Fundamental Theorem of
Arithmetic
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Lemma 4.2: If a, b  Z+ and p is prime, then
p|ab  p|a or p|b.
Proof: See page 207.
Lemma 4.3:
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Let ai  Z+ for all 1≦i≦n. If p is prime and
p|a1a2…an, then p|ai for some 1≦i≦n.
Example 4.38
Theorem 4.11:
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Every integer n > 1 can be written as a
product of primes uniquely, up to the
order of the primes. (Here a single prime
is considered a product of one factor.)
Example 4.39: page 208.