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**Exercise Set 4.1**
In 1-4 justify your answers by using the definitions of even, odd, prime, and
composite numbers.
1. Assume that $k$ is a particular integer.
a. Is $-17$ an odd integer?
b. Is $0$ neither even nor odd?
c. Is $2k - 1$ odd?
2. Assume that $c$ is a particular integer.
a. Is $-6c$ an even integer?
b. Is $8c + 5$ an odd integer?
c. Is $(c^1 + 1) - (c^2 - 1) - 2$ an even integer?
3. Assume that $m$ and $n$ are particular integers?
a. Is $6m + 8n$ even?
b. Is $10mn + 7$ odd?
c. If $m > n > 0$, is $m^2 - n^2$ composite?
4. Assume that $r$ and $s$ are particular integers.
a. Is $4rs$ even?
b. Is $6r + 4s^2 + 3$ odd?
c. If $r$ and $s$ are both positive, is $r^2 + 2rs + s^2$ composite?
Prove the statements in 5-11.
5. There are integers $m$ and $n$ such that $m > 1$ and $n > 1$ and
$\dfrac{1}{m} + \dfrac{1}{n}$ is an integer.
6. There are distinct integers $m$ and $n$ such that
$\dfrac{1}{m} + \dfrac{1}{n}$ is an integer.
7. There are real numbers $a$ and $b$ such that
$$ \sqrt{a + b} = \sqrt{a} + \sqrt{b} $$
8. There is an integer $n > 5$ such that $2^n - 1$ is prime.
9. There is a real number $x$ such that $x > 1$ and $2^x > x^{10}$.
**Definition:** An integer $n$ is called a **perfect square** if, and only if,
$n = k^2$ for some integer $k$.
10. There is a perfect square that can be written as a sum of two other perfect
squares.
11. There is an integer $n$ such that $2n^2 - 5n + 2$ is prime.
In 12-13, (a) write a negation for the given statement, and (b) use a
counterexample to disprove the given statement. Explain how the counterexample
actually shows that the given statement is false.
12. For all real numbers $a$ and $b$, if $a < b$ the $a^2 < b^2$.
13. For every integer $n$, if $n$ is odd then $\dfrac{n - 1}{2}$ is odd.
Disprove each of the statements in 14-16 by giving a counterexample. In each
case explain how the counterexample actually disproves the statement.
14. For all integers $m$ and $n$, if $2m + n$ is odd then $m$ and $n$ are both
odd.
15. For every integer $p$, if $p$ is prime then $p^2 - 1$ is even.
16. For every integer $n$, if $n$ is even then $n^2 + 1$ is prime.
In 17-20, determine whether the property is true for all integers, true for no
integers, or true for some integers and false for other integers. Justify your
answers.
17. $(a + b)^2 = a^2 + b^2$
18. $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a + c}{b + d}$
19. $-a^n = (-a)^n$
20. The average of any two odd integers is odd.
Prove the statement in 21 and 22 by the method of exhaustion.
21. Every positive even integer less than 26 can be expressed as a sum of three
of fewer perfect squares. (For instance, $10 = 1^2 + 3^2$ and $16 = 4^2$.)
22. For each integer $n$ with $1 \leq n \leq 10$, $n^2 -n + 11$ is a prime
number.
Each of the statements in 23-26 is true. For each, (a) rewrite the statement
with the quantification implicit as If _____, then _____, and (b) write the
first sentence of a proof (the "starting point") and the last sentence of a
proof (the "conclusion to be shown"). (Note that you do not need to understand
the statements in order to be able to do these exercises.)
23. For every integer $m$, if $m > 1$ then $0 < \dfrac{1}{m} < 1$.
24. For every real number $x$, if $x > 1$ then $x^2 > x$.
25. For all integers $m$ and $n$, if $mn = 1$ then $m = n = 1$ or $m = n = -1$.
26. For every real number $x$, if $0 < x < 1$ then $x^2 < x$.
27. Fill in the blanks in the following proof.
**Theorem:** For every odd integer $n$, $n^2$ is odd.
**Proof:** Suppose $n$ is any ___ (a) ___. By definition of odd, $n = 2k + 1$
for some integer $k$. Then
$$ n^2 = \left(___(b)____\right)^2 \quad \text{ by substitution} $$
$$ \quad = 4k^2 + 4k + 1 \quad \text{ by multiplying out} $$
$$ \quad = 2(2k^2 + 2k) + 1 \quad \text{ by factoring out a 2} $$
Now $2k^2 + 2k$ is an integer because it is a sum of products of integers.
Therefore $n^2$ equals $2 \cdot (\text{an integer}) + 1$, and so ___ (c) ___ is
odd by definition of odd.
Because we have not assumed anything about $n$ except that it is an odd integer,
it follows from the principle of ___ (d) ___ that for _every_ odd integer $n$,
$n^2$ is odd.
In each of 28-31:
a. Rewrite the theorem in three different ways:
as $\forall$ _____, if _____ then _____, as $\forall$ _____, _____ (without
using the words _if_ or _then_),
and as If _____, then _____ (without using an explicit universal quantifier).
b. Fill in the blanks in the proof of the theorem.
28.
**Theorem:** the sum of any two odd integers is even.
**Proof:** Suppose $m$ and $n$ are any _[particular but arbitrarily chosen]_ odd
integers.
_[We must show that $m + n$ is even.]_
By __ (a) __, $m = 2r + 1$ and $n = 2s + 1$ for some integers $r$ and $s$.
Then
$$ m + n = (2r + 1) + (2s + 1) \quad \text{k by \_\_ (b) \_\_} $$
$$ \quad = 2r + 2s + 2 $$
$$ \quad = 2(r + s + 1) \quad \text{ by algebra} $$
Let $u = r + s + 1$. Then $u$ is an integer because $r$, $s$, and $1$ are
integers and because __ \(c\) __.
Hence $m + n = 2u$, where $u$ is an integer, and so, by __ (d) __, $m + n$ is
even _[as was to be shown]._
29.
**Theorem:** The negative of any integer is even.
**Proof:** Suppose $n$ is any _[particular but arbitrarily chosen]_ even
integer.
_[We must show that $-n$ is even.]_
By __ (a) __, $n = 2k$ for some integer $k$.
Then
$$ -n = -(2k) \quad \text{ by \_\_ (b) \_\_} $$
$$ \quad = 2(-k) \quad \text{ by algebra} $$
Let $r = -k$. Then $r$ is an integer because $(-1)$ and $k$ are integers and __
\(c\) __.
Hence $-n = 2r$, where $r$ is an integer, and so $-n$ is even by __ (d) __ _[as
was to be shown]._
30.
**Theorem 4.1.2:** The sum of any even integer and any odd integer is odd.
**Proof:** Suppose $m$ 8s any even integer and $n$ is __ (a) __. By definition
of even, $m = 2$ for some __ (b) __, and by definition of odd, $n = 2s + 1$ for
some integer $s$. By substitution and algebra,
$$ m + n = \text{\_\_ (c) \_\_} = 2(r + s) + 1 $$
Since $r$ and $s$ are both integers, so is their sum $r + s$. Hence $m + n$ has
the form twice some integer plus one, and so __ (d) __ by definition of odd.
31.
**Theorem:** Whenever $n$ is an odd integer, $5n^2 + 7$ is even.
**Proof:** Suppose $n$ is any _[particular but arbitrarily chosen]_ odd integer.
_[We must show that $5n^2 + 7$ is even.]_
By definition of odd, $n$ = __ (a) __ for some integer $k$.
Then
$$ 5n^2 + 7 = \text{\_\_ (b) \_\_} \quad \text{ by substitution} $$
$$ \quad = 5(4k^2 + 4k + 1) + 7 $$
$$ \quad = 20k^2 + 20k + 12 $$
$$ \quad = 2(10k^2 + 10k + 6) \quad \text{ by algebra} $$
Let $t =$ __ \(c\) __. Then $t$ is an integer because products and sums of
integers are integers.
Hence $5n^2 + 7 = 2t$, where $t$ is an integer, and thus __ (d) __ by definition
of even _[as was to be shown]._

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**Assumptions**
- In this text we assume familiarity with the laws of basic algebra, which are
listed in Appendix A.
- We also use the three properties of equality: For all objects $A$, $B$, and
$C$, (1) $A = A$, (2) if $A = B$, then $B = 1$, and (3) if $A = B$ and
$B = C$, then $A = C$.
- And we use the principle of substitution: For all objects $A$ and $B$, if
$A = B$, then we may substitute $B$ whenever we have $A$.
- In addition, we assume that there is no integer between $0$ and $1$ and that
the set of all integers is closed under addition, subtraction, and
multiplication. This means that sums, differences, and products of integers
are integers.
---
Page 185
**Definitions**
An integer $n$ is **even** if, and only if, $n$ equals twice some integer. An
integer $n$ is **odd** if, and only if, $n$ equals twice some integer plus $1$.
Symbolically, for any integer $n$
$$ n \text{ is even} \Leftrightarrow n = 2k \text{ for some integer } k $$
$$ n \text{ is odd} \Leftrightarrow n = 2k + 1 \text{ for some integer } k $$
---
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**Definition**
An integer $n$ is **prime** if, and only if, $n > 1$ and for all positive
integers $r$ and $s$, if $n = rs$, then either $r$ or $s$ equals $n$. An integer
$n$ is **composite** if, and only if, $n > 1$ and $n = rs$ for some integers $r$
and $s$ with $1 < r < n$ and $1 < s < n$.
In symbols: For each integer $n$ with $n > 1$,
$$ n \text{ is prime} \Leftrightarrow \forall \text{ positive integers } r \text{ and } s, \text{ if } n = rs \text{ then either } r = 1 \text{ and } s = n \text{ or } r = n \text{ and } s = 1 $$
$$ n \text{ is composite} \Leftrightarrow \exists \text{ positive integers } r \text{ and } s \text{ such that } n = rs \text{ and } 1 < r < n \text{ and } 1 < s < n $$
---
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**Disproof by Counterexample**
To disprove a statement of the form
"$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$," find a value of $x$ in
$D$ for which the hypothesis $P(x)$ is true and the conclusion $Q(x)$ is false.
Such an $x$ is called a **counterexample**.
---
Page 189
**Generalizing from the Generic Particular**
To show that _every_ element of a set satisfies a certain property, suppose $x$
is a _particular_ but _arbitrarily chosen_ element of the set, and show that $x$
satisfies the property.
---
Page 191
**Existential Instantiation**
If the existence of a certain kind of object is assumed or has been deduce, then
it can be given a name, as long as that name is not currently being used to
refer to something else in the same discussion.
---
Page 192
**Theorem 4.1.1**
The sum of any two even integers is even.
**Proof:** Suppose $m$ and $n$ are any _[particular but arbitrarily chosen]_
even integers. _[We must show that $m + n$ is even.]_ By definition of even,
$m = 2r$ and $n = 2s$ for some integers $r$ and $s$. Then
$$ m + n = 2r + 2s \quad \text{ by substitution} $$
$$ \quad = 2(r + s) \quad \text{ by factoring out a 2} $$
Let $t = r + s$. Note that $t$ is an integer because it is a sum of integers.
Hence
$$ m + n = 2r \quad \text{where } t \text{ is an integer} $$
It follows by definition of even that $m + n$ is even. _[This is what we needed
to show.]_

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**Test Yourself**
Page 194
1. An integer is even if, and only if, ______.
2. An integer is odd if, and only if, ______.
3. An integer $n$ is prime if, and only if, ______.
4. The most common way to disprove a universal statement is to find ______.
5. According to the method of generalizing from the generic particular, to show
that every element of a set satisfies a certain property, suppose $x$ is a
______, and show that ______.
6. To use the method of direct proof to prove a statement of the form, "For
every $x$ in a set $D$, if $P(x)$ then $Q(x)$," one supposes that ______ and
one shows that ______.