For example, a well-known function due to Leonhard Euler is deﬁned by the
formula
f
(
n
) =
n
2
+
n
+
41 for all integers
n
. If you substitute the numbers
n
=
0
,
1
,
2
,...,
39 into this function, you obtain the numbers 41, 43, 47,
...
, 1601, all
of which are prime numbers. It therefore might appear that substituting in every
positive integer into this function would result in a prime number (which would be
a very nice property), but it turns out that
f
(
40
) =
1681
=
41
2
, which is not prime.
See [Rib96, p. 199] for more discussion of this, and related, functions. The point is
that if you want to prove that a statement is true for all
x
in
U
, it does not sufﬁce to
try only some of the possible values of
x
.
Statements of the form
(
∀
x
in
U
)
P
(
x
)
can be proved by strategies other than di-
rect proof. For example, the proof of such a statement using proof by contradiction
typically has the following form.
Proof.
We use proof by contradiction. Let
y
0
be in
U
. Suppose that
P
(
y
0
)
is false.
.
.
.
(argumentation)
.
.
.
Then we arrive at a contradiction.
u
t

72
2 Strategies for Proofs
We will not show here any examples of proofs of statements of the form
(
∀
x
in
U
)
P
(
x
)
, because we have already seen a number of such proofs in the pre-
vious sections of this chapter.
We now consider statements with a single existential quantiﬁer, that is, statements
of the form “
(
∃
x
in
U
)
P
(
x
)
.” Using the Existential Generalization rule of inference in
Section 1.5, we see that to prove a theorem of the form
(
∃
x
)
P
(
x
)
means that we need
to ﬁnd some
z
0
in
U
such that
P
(
z
0
)
holds. It does not matter if there are actually
many
x
in
U
such that
P
(
x
)
holds; we need to produce only one of them to prove
existence. A proof of “
(
∃
x
in
U
)
P
(
x
)
” can also be viewed as involving a statement
of the form
A
→
B
. After we produce the desired object
z
0
in
U
, we then prove the
statement “if
x
=
z
0
, then
P
(
x
)
is true.” Such a proof typically has the following form.
Proof.
Let
z
0
=
...
.
.
.
.
(argumentation)
.
.
.
Then
z
0
is in
U
.
.
.
.
(argumentation)
.
.
.
Then
P
(
z
0
)
is true.
u
t
How we ﬁnd the element
z
0
in the above type of proof is often of great interest,
and sometimes is the bulk of the effort we spend in ﬁguring out the proof, but it is
not part of the actual proof itself. We do not need to explain how we found
z
0
in
the ﬁnal write-up of the proof. The proof consists only of deﬁning
z
0
, and showing
that
z
0
is in
U
, and that
P
(
z
0
)
is true. It is often the case that we ﬁnd
z
0
by going
backwards, that is, assuming that
P
(
z
0
)
is true, and seeing what
z
0
has to be. However,
this backwards work is not the same as the actual proof, because, as we shall see,
not all mathematical arguments can be reversed—what works backwards does not
necessarily work forwards.
We now turn to a simple example of a proof involving an existential quantiﬁer.
Recall the deﬁnitions concerning 2
×
2 matrices prior to Theorem 2.4.5. We say that
a 2
×
2 matrix
M
=
(
a b
c d
)
has integer entries if
a
,
b
,
c
and
d
are integers.