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infinite in both directions can do. P1: 1
CH137T Rosen2311T MHIA017Rosenv5.cls May 13, 2011 10:27 904
P1: 1 APP17T Rosen2311T MHIA017Rosenv5.cls May 13, 2011 10:28 1
APPENDIX Axioms for the Real
Numbers and the Positive Integers In
this book we have
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statements are axioms, they are
commonly called laws or rules. The
first two of these axioms tell us that
when we add or multiply two real
numbers, the result is again a real
number; these are the closure laws.
Closure law for addition For all real
number
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calculated an approximation of .
Archimedes was also an accomplished
engineer and inventor; his machine for
pumping water, now called
Archimedes screw, is still in use today.
Perhaps his best known discovery is
the principle of buoyancy, which tells
us th
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rational number (that is, a number of
the form x/y, where x and y are
integers with y = 0). Exercises 21 and
22 involve the notion of an equivalence
relation, discussed in Chapter 9 of the
text. 21. Define a relation on the
set of ordered pairs of positiv
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material should consult precalculus or
calculus books, such as those
mentioned in the Suggested Readings.
Exponential Functions Let n be a
positive integer, and let b be a fixed
positive real number. The function
fb(n) = bn is defined by fb(n) = bn = b
b
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These laws are called identity laws.
Additive identity law For every real
number x, x + 0 = 0 + x = x.
Multiplicative identity law For every
real number x, x 1 = 1 x = x. Although
it seems obvious, we also need the
following axiom. Identity elements
axiom
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Often, we require the use of a more
general type of construction. This is
used when we wish to do one thing
when the indicated condition is true,
but another when it is false. We use the
construction if condition then
statement 1 else statement 2 Note tha
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real number x there exists an integer n
such that n>x. Proof: Suppose that x is
a real number such that n x for every
integer n. Then x is an upper bound of
the set of integers. By the
completeness property it follows that
the set of integers has a least
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Projects Respond to these with essays
using outside sources. 1. Describe how
the growth of certain types of plants
can be modeled using a Lidenmeyer
system. Such a system uses a grammar
with productions modeling the
different ways plants can grow. 2.
Desc
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this last inequality becomes 1 > 0,
contradicting the trichotomy law
because we had assumed that 0 > 1.
Because we know that 0 = 1 and that it
is impossible for 0 > 1, by the
trichotomy law, we conclude that 1 > 0.
ARCHIMEDES (287 b.c.e.212 b.c.e.)
Archim
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same result when we first add a pair of
real numbers and then multiply by a
third real number or when we multiply
each of these two real numbers by the
third real number and then add the
two products. Distributive laws For all
real numbers x, y, and z, x
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equivalence classes are discussed in
Chapter 9.) Next, the set of rational
numbers can be constructed using the
equivalence classes of pairs of integers
where the second integer in the pair is
not zero, where (a, b) (c, d) if and
only if a d = b c; additi
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[Sm98]. Prolog is discussed in depth in
Nilsson and Maluszynski [NiMa95] and
in Clocksin and Mellish [ClMe94]. The
basics of proofs are covered in
Cupillari [Cu05], Morash [Mo91], Solow
[So09], Velleman [Ve06], and Wolf
[Wo98]. The science and art of
cons
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of statements. P1: 1 APP37T Rosen2311T MHIA017Rosenv5.cls May 13,
2011 10:28 A14 Appendix 3 /
Pseudocode Loop Constructions There
are two types of loop construction in
the pseudocode in this book. The first
is the for construction, which has the
form
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Mathematical induction axiom If S is a
set of positive integers such that 1 S
and for all positive integers n if n S,
then n + 1 S, then S is the set of
positive integers. Most mathematicians
take the real number system as already
existing, with the real
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statement the lefthand side is the
name of the variable and the righthand side is an expression that
involves constants, variables that have
been assigned values, or functions
defined by procedures. The righthand
side may contain any of the usual
arithm
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a greater than relationship, the greater
than relationship is preserved and
when we multiply both sides of a
greater than relationship by a positive
real number (that is, a real number x
with x > 0), the greater than
relationship is preserved. Additive
co
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by qj is qj 1. All odd primes other than
3 are of the form 6k + 1 or 6k + 5, and
the product of primes of the form 6k + 1
is also of this form. Therefore at least
one of the pis must be of the form 6k+5,
a contradiction. 31. The product of
numbers of the
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Prove that for all real numbers x, y, and
z, if x + z = y + z, then x = y. 7. Prove that
for every real number x, (x) = x.
Define the difference x y of real
numbers x and y by x y = x + (y),
where y is the additive inverse of y,
and the quotient x/y, wher
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your answer. 10. Define a linearbounded automaton. Explain how
linearbounded automata are used to
recognize sets. Which sets are
recognized by linearbounded
automata? Provide an outline of a
proof justifying your answer. 11. Look
up Turings original defi
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more general for statement, of the
form for all elements with a certain
property is used in this text. This
means that the statement or block of
statements that follow are carried out
successively for the elements with the
given property. The second type
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function has an inverse logb x, called
the logarithmic function to the base b.
In other words, if b is a real number
greater than 1 and x is a positive real
number, then blogb x = x. The value of
this function at x is called the
logarithm of x to the base
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pseudocode used in the text. This
pseudocode is designed so that its
basic structure resembles that of
commonly used programming
languages, such as C+ and Java, which
are currently the most commonly
taught programming languages.
However, the pseudocode we
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Exercises the sum of a multiple of 2 (by
the inductive hypothesis) and a
multiple of 2 (by definition), hence,
divisible by 2. 33. Let P (n) be n5 n is
divisible by 5. Basis step: P (0) is true
because 05 0 = 0 is divisible by 5.
Inductive step: Assume th
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Knuth on the art of computer
programming [Kn97a], [Kn97b], and
[Kn98]; Kronsj [Kr87]; Levitin [Le06];
Manber [Ma89]; Pohl and Shaw
[PoSh81]; Purdom and Brown
[PuBr85]; Rawlins [Ra92]; Sedgewick
[Se03]; Wilf [Wi02]; and Wirth [Wi76].
Sorting and searching
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the cases when i + j 2. The knight is
already at (0, 0) to start, so the empty
sequence of moves reaches that square.
To reach (1, 0), the knight moves (0, 0)
(2, 1) (0, 2) (1, 0). Similarly, to
reach (0, 1), the knight moves (0, 0)
(1, 2) (2, 0) (0, 1)
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3(k + 1) 1]. This demonstrates that P
(k + 1) is true. 19. a) 1 + 1 4 < 2 1 2 b)
This is true because 5/4 is less than
6/4. c) 1+ 1 4 + 1 k2 < 2 1 k d) For
each k 2 that P (k) implies P (k +1); in
other words, we want to show that
assuming the inductive h