out the remainder of the proof, but leave
the justification of these steps until
Chapter 4. Because 3 is a factor of c2, it
must also be a factor of c. Furthermore,
because 3 is a factor of c, 9 is a factor of
c2, which means that 9 is a factor of 3d2.
Th

proof of a conditional statement, you start
with the premises. Using these premises,
together with axioms and known
theorems, you can construct a proof using
a sequence of steps that leads to the
conclusion. This type of reasoning, called
forward reasonin

CH01-7T Rosen-2311T MHIA017-Rosenv5.cls May 13, 2011 15:27 1.8 Proof
Methods and Strategy 103 FIGURE 2 The
Standard Checkerboard. FIGURE 3 Two
Dominoes. the development of new parts
of mathematics. We will mention a few
famous conjectures later in this se

mean is always greater than the geometric
mean. [For example, when x = 4 and y = 6,
we have 5 = (4 + 6)/2 > 4 6 = 24.] Can
we prove that this inequality is always
true? Solution: To prove that (x + y)/2 >
xy when x and y are distinct positive real
numbers

= 0 whenever n is an integer with n > 2.
Remark: The equation x2 + y2 = z2 has
infinitely many solutions in integers x, y,
and z; these solutions are called
Pythagorean triples and correspond to
the lengths of the sides of right triangles
with integer len

some element of the domain. Functions
with this property are called onto
functions. DEFINITION 7 A function f from
A to B is called onto, or a surjection, if and
only if for every element b B there is an
element a A with f (a) = b. A function f
is called

EXAMPLE 22 Let g be the function from
the set cfw_a, b, c to itself such that g(a) = b,
g(b) = c, and g(c) = a. Let f be the function
from the set cfw_a, b, c to the set cfw_1, 2, 3
such that f (a) = 3, f (b) = 2, and f (c) = 1.
What is the composition of

one sequence is sorted into
nondecreasing order and the other is
sorted into nonincreasing order. 38. Prove
or disprove that if you have an 8-gallon
jug of water and two empty jugs with
capacities of 5 gallons and 3 gallons,
respectively, then you can mea

T1: 2 CH01-7T Rosen-2311T MHIA017Rosen-v5.cls May 13, 2011 15:27 1.8
Proof Methods and Strategy 105 squares.
We can use these observations to prove by
contradiction that a standard
checkerboard with opposite corners
removed cannot be tiled using dominoes.

their proofs. Such presentations do not
convey the discovery process in
mathematics. This process begins with
exploring concepts and examples, asking
questions, formulating conjectures, and
attempting to settle these conjectures
either by proof or by coun

conjecture, sometimes known as the 3x +
1 conjecture, states that for all positive
integers x, when we repeatedly apply the
transformation T , we will eventually
reach the integer 1. For example, starting
with x = 13, we find T (13) = 3 13 + 1 =
40, T (40

onto but not one-to-one, the third is both
one-to-one and onto, and the fourth is
neither one-to-one nor onto. The fifth
correspondence in Figure 5 is not a
function, because it sends an element to
two different elements. Suppose that f is a
function from

Section 1.8, we will develop a large
arsenal of proof techniques that can be
used to prove a wide variety of theorems.
When you read proofs, you will often find
the words obviously or clearly. These
words indicate that steps have been
omitted that the aut

years, several established mathematicians
thought that they had proved this
theorem. In the nineteenth century, one of
these failed attempts led to the
development of the part of number theory
called algebraic number theory. A correct
P1: 1/1 P2: 1/2 QC:

square is colored blue. Therefore, we
assume that the remaining board contains
20 blue squares, 21 black squares, and 22
white squares. If we could tile this board
using straight triominoes, then we would
use 63/3 = 21 straight triominoes. These
triominoe

numbers) and its codomain is the set of
integers. A function is called realvalued if its codomain is the set of real
numbers, and it is called integer-valued if
its codomain is the set of integers. Two
real-valued functions or two
integervalued functions

to find a counterexample, first by looking
at the simplest, smallest examples. If you
cannot find a counterexample, you might
again try to prove the statement. In any
case, looking for counterexamples is an
extremely important pursuit, which often
provide

universe of discourse is the domain of the
function. We illustrate this concept by
giving examples of functions that are oneto-one and other functions that are not
one-to-one. EXAMPLE 8 Determine
whether the function f from cfw_a, b, c, d to
cfw_1, 2, 3,

= r, then as + b = 0. This establishes the
uniqueness part of the proof. P1: 1/1
P2: 1/2 QC: 1/1 T1: 2 CH01-7T Rosen2311T MHIA017-Rosen-v5.cls May 13,
2011 15:27 100 1 / The Foundations:
Logic and Proofs Proof Strategies Finding
proofs can be a challengin

reach a dead end however we
successively place dominoes on the board.
To construct such a proof, we would have
to consider all possible cases that arise as
we run through all possible choices of
successively placing dominoes. For
example, we have two choi

such that n2 + n3 = 100. 30. Prove that
there are no solutions in integers x and y
to the equation 2x2 + 5y2 = 14. 31. Prove
that there are no solutions in positive
integers x and y to the equation x4 + y4 =
625. 32. Prove that there are infinitely
many s

method, which can be used to prove
results about the size of infinite sets, in
Section 2.5. In Chapter 6 we will introduce
the notion of combinatorial proofs, which
can be used to prove results by counting
arguments. The reader should note that
entire boo

employees is assigned a job from a set of
possible jobs, each to be done by a single
worker. In this situation, the function f
that assigns a job to each worker is oneto-one. To see this, note that if x and y are
two different workers, then f (x) = f (y)

= y. Once we have carried out this
backward reasoning, we can easily
reverse the steps to construct a proof
using forward reasoning. We now give
this proof. Suppose that x and y are
distinct positive real numbers. Then (x
y)2 > 0 because the square of a

c d 1 2 3 4 a b c 1 2 3 4 One-to-one, not
onto Onto, not one-to-one One-to-one,
and onto Neither one-to-one nor onto (a)
(b) (c) (d) (e) Not a function FIGURE 5
Examples of Different Types of
Correspondences. Solution: This function
is onto, because for e

1 exists and is a one-to-one
correspondence from B to A. The inverse
function reverses the correspondence of
the original function, so f 1(b) = a when f
(a) = b, and f (a) = b when f 1(b) = a.
Hence, (f 1 f )(a) = f 1(f (a) = f 1(b) =
a, and (f f 1)(b) =

Display the graph of the function f (n) =
2n + 1 from the set of integers to the set of
integers. Solution: The graph of f is the set
of ordered pairs of the form (n, 2n + 1),
where n is an integer. This graph is
displayed in Figure 8. EXAMPLE 25
Display

can always win the game, we work
backward. At the last step, the first player
can win if this player is left with a pile
containing one, two, or three stones. The
second player will be forced to leave one,
two, or three stones if this player has to
remove

constructive or nonconstructive? 10.
Prove that either 2 10500 + 15 or 2
10500 + 16 is not a perfect square. Is your
proof constructive or nonconstructive?
11. Prove that there exists a pair of
consecutive integers such that one of
these integers is a pe

triominoes to tile a standard
checkerboard with one of its four corners
removed? An 8 8 checkerboard with one
corner removed contains 64 1 = 63
squares. Any tiling by straight triominoes
of one of these four boards uses 63/3 = 21
triominoes. However, when