b)
We can use calculus to show that the minimum value of
ݔ
ଶ
+2ݔ+2
is
1
. Let
݂ሺݔሻ=ݔ
ଶ
+2ݔ+2
Then
݂
ᇱ
ሺݔሻ=2ݔ+2
and
݂
ᇱ
ሺݔሻ=0
when
ݔ =−1
. Since
݂
ᇱᇱ
ሺݔሻ=2
, which is positive,
the second derivative test tells is that
ݔ =−1
is a minimum. The minimum value of
݂
is
݂ሺ−1ሻ=ሺ−1ሻ
ଶ
+2ሺ−1ሻ+2=1
, so
݂ሺݔሻ≥1
and
݂ሺݔሻ≠0
for any real number
ݔ
.
c)
If we plot the curve
ݕ =ݔ
ଶ
+2ݔ+2
, we find that the curve does not intersect the
ݔ
-axis,
signifying that
ݕ ≠0
for any
ݔ
.
This example shows us that in the actual carrying out a mathematical proof, you need “mathematical
knowledge”
in addition to
knowing the structure or what it means to give a formal, rigorous proof. In this
course, we focus on proof techniques, but we also do not shy away from alluding to or including relevant
“mathematical knowledge” to enrich our study.

12
4.
Definitions, Theorems, and Proofs
We have introduced the basic elements of mathematical statements and arguments, and have seen some
examples of proofs. In this course we also hope to learn to appreciate the fact that there are many ways
of
writing a correct proof, and of course more ways of writing a wrong proof! Even among correct proofs of the
same theorem, you may “like” some proofs more than the others. In any case, our goal is two-fold: to write a
proof that is clear and “easy” to follow and to understand a well-written proof. In this section, we will present
examples of writing a proof in different ways. First let us gather all the definitions we may use.
Definition 1
An integer
ݔ
is
even
if
ݔ =2݊
for some integer
݊
.
Definition 2
An integer
ݔ
is
odd
if
ݔ =2݊+1
for some integer
݊
.
Definition 3
An integer
ܽ
is
divisible
by an integer
ܾ
(denoted
ܾ|ܽ
), if
ܾܿ =ܽ
for some integer
ܿ
. We
also say “
ܾ
divides
ܽ
”, “
ܾ
is a
factor
of
ܽ
”, “
ܾ
is a
divisor
of
ܽ
”, or “
ܽ
is a
multiple
of
ܾ
”.
Definition 4
An integer
is
prime
if
>1
and the only positive divisors of
are
1
and
.
Definition 5
An integer is
composite
if there is an integer
ܾ
such that
ܾ|ܽ
and
1<ܾ <ܽ
.
Example 1
Prove the following proposition.
Proposition
An integer
ݔ
is odd if and only if
ݔ+1
is even.
Proof 1
The proposition is a biconditional, and we need to prove in two directions.
(
⇒
)
We show that if
ݔ
is odd then
ݔ+1
is even:
ݔ
is odd
→
ݔ =2݊+1
for some integer
݊
(Definition 2)
→
ݔ+1=2݊+1+1=2݊+2=2ሺ݊+1ሻ
→
ݔ+1=2݇
, where
݇ =݊+1
is an integer
→
ݔ+1
is even
(Definition 1)
(
⇐
)
We show that if
ݔ+1
is even, then
ݔ
is odd:
ݔ+1
is even
→
ݔ+1=2݊
for some integer
݊
(Definition 1)
→
ݔ =2݊−1=2ሺ݊−1ሻ+1
, where
݊−1
is an integer
→
ݔ =2݇+1
, where
݇ =݊−1
is an integer
→
ݔ
is odd
(Definition 2)
■
Remarks:
•
The symbol
■
signifies the end of a proof.

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