1.
If \(n\) is even then \(n^2+3n+5\) is odd.
Exercises 3.5 Exercises
2.
Prove that the product of two odd numbers is odd.
3.
We have already seen a proof that the product of two odd numbers is also odd. We’ll now look at the remaining cases for the parity of a product or sum of two integers.
For each of the following cases, determine if the resulting number is even or odd, and prove your statement:
- the sum of two odd numbers;
- the sum of two even numbers;
- the sum of an even and an odd number;
- the product of two even numbers;
- the product of an even and an odd number.
4.
Consider the faulty proof below for the following statement:
\begin{gather*}
\text{Show that if } x+y \text{ is odd, then either } x \text{ or } y \text{ is odd, but not both}.
\end{gather*}
Faulty proof.
Assume that either \(x\) or \(y\) is odd, but not both. Assume that \(x\) is odd and \(y\) is even (otherwise, switch \(x\) and \(y\) in the following argument). By the definitions of odd and even numbers, we know that \(x=2n+1\) and \(y=2m\) for some \(n,m \in \mathbb{Z}\text{.}\) Then
\begin{equation*}
x + y = (2n+1) + (2m)=2 (n+m) + 1.
\end{equation*}
Since \(n, m \in \mathbb{Z}\) and the sum of integers is also an integer, we see that \(n+m\in \mathbb{Z}\text{,}\) so that \(x+y\) fits the definition of an odd number.
Identify any issues with the proof as written above.
5.
Consider the faulty proof below for the following statement:
The sum of two odd integers is even.
Faulty proof.
Given \(a=2k+1\) and \(b=2\ell+1\text{,}\)
\begin{equation*}
a+b = (2k+1)+(2\ell+1)=2(k+\ell+1).
\end{equation*}
Since \(k+\ell+1\in \mathbb{Z}\text{,}\) \(a+b\) is even.
Identify any issues with the proof as written above, and then give a proper proof of the statement.
6.
Let \(n,a,b,x,y\in\mathbb{Z}\text{.}\) If \(n\mid a\) and \(n\mid b\text{,}\) then \(n\mid (ax+by)\text{.}\)
7.
Let \(n, a\in\mathbb{Z}\text{.}\) Prove that if \(n\mid a\) and \(n\mid (a+1)\text{,}\) then \(n=-1\) or \(n=1\text{.}\)
8.
Let \(a\in \mathbb{Z}\text{.}\) If \(3\mid a\) and \(2\mid a\text{,}\) then \(6\mid a\text{.}\)
9.
Let \(n\in\mathbb{Z}\text{.}\) If \(3\mid (n-4)\text{,}\) then \(3\mid (n^2-1)\text{.}\)
10.
Consider the faulty proof below for the following statement:
Let \(a\text{,}\) \(b\text{,}\) and \(c\) be integers. If \(a\mid b\) and \(b\mid c\text{,}\) then \(a\mid c\text{.}\)
Faulty proof.
Assume \(a\text{,}\) \(b\text{,}\) and \(c\) are integers such that \(a\mid b\) and \(b\mid c\text{.}\) Since \(a\) divides \(b\text{,}\) we have that \(b=ka\) for some \(k\in\mathbb{Z}\text{.}\) Moreover, since \(b\) divides \(c\text{,}\) we have that \(c=kb\) for some \(k\in\mathbb{Z}\text{.}\) But then
\begin{equation*}
c=k(ka)=k^2a,
\end{equation*}
Since \(k\) is an integer, \(k^2\in\mathbb{Z}\text{,}\) and it follows that \(a\) divides \(c\text{.}\)
Identify any issues with the proof as written above, and then give a correct proof of the statement.
11.
Consider the faulty proof that \(2=1\text{.}\)
Faulty proof.
Assume that \(x=y\text{.}\) Then multiplying both sides by \(x\) gives
\begin{align*}
x^2 \amp = xy\\
\Rightarrow x^2-y^2 \amp = xy - y^2\\
\Rightarrow (x+y)(x-y) \amp = y(x-y)\\
\Rightarrow x+y \amp = y\\
\Rightarrow 2y \amp = y
\end{align*}
Letting \(x=y=1\text{,}\) we have shown that \(2=1\text{.}\)
Identify any issues with the proof as written above.
12.
The floor function, denoted by \(\lfloor x \rfloor \text{,}\) is defined to be the function that takes a real number \(x\) and returns the greatest integer less than or equal to \(x\text{.}\) This is also sometimes called the greatest integer function. For example,
\begin{equation*}
\lfloor 3.5 \rfloor=3,
\qquad
\lfloor -2.5 \rfloor=-3,
\qquad \text{and} \qquad
\lfloor 7 \rfloor=7.
\end{equation*}
Using this definition, prove that
\begin{equation*}
\lfloor x \rfloor=x
\implies
x\in\mathbb{Z}\text{,}
\end{equation*}
and that
\begin{equation*}
x\in\mathbb{Z}
\implies
\lfloor x \rfloor=x.
\end{equation*}
13.
Definition: We call a number \(n\) an integer root if \(n^k=m\) for some \(k\in \mathbb{N}\) and \(m \in \mathbb{Z}\text{.}\)
For example, \(\sqrt{7}\) is an integer root because \(\left( \sqrt{7}\right)^2 = 7\text{.}\) However, \(\frac{5}{3}\) is not an integer root (but proving that is a little beyond this point in the text).
Use the above definition to show that if \(a\) and \(b\) are integer roots, then so is \(ab\text{.}\)
14.
Consider the faulty proof below for the following statement:
Let \(x\) be a positive real number. If \(x \lt 1\text{,}\) then \(1 \lt \dfrac{3x+2}{5x}\text{.}\)
Faulty proof.
Let \(x\) be positive. Then by multiplying the inequality
\begin{equation*}
1 \lt \frac{3x+2}{5x}
\end{equation*}
by \(5x\text{,}\) which is positive, we obtain
\begin{equation*}
5x \lt 3x+2 .
\end{equation*}
Collecting like terms, we have \(2x \lt 2\text{,}\) and finally dividing by \(2\text{,}\) we have \(x \lt 1\text{.}\)
Identify any issues with the proof as written above, and then give a correct proof of the statement.
15.
Consider the faulty proof below for the following statement:
Let \(x\) be a negative real number. Show that \(-1 \lt \dfrac{5}{3x-5}\text{.}\)
Faulty proof.
Let \(x\) be negative. Then by multiplying the inequality
\begin{equation*}
-1 \lt \dfrac{5}{3x-5}
\end{equation*}
by \(3x-5\) we obtain
\begin{equation*}
-3x+5 \lt 5 .
\end{equation*}
and therefore \(-3x \lt 0\text{.}\) Dividing by \(-3\) we end up with \(x \lt 0\text{,}\) which is true.
Identify any issues with the proof as written above, and then give a correct proof of the statement.
16.
Let \(x,y\) be positive real numbers. Without using Calculus, prove that
\begin{equation*}
(x \gt y) \implies (\sqrt{x} \gt \sqrt{y})
\end{equation*}
17.
Consider the faulty proof below for the following statement:
Let \(a,b\in \mathbb{R}\text{.}\) If \(0 \lt a \lt b\text{,}\) then
\begin{equation*}
\sqrt{ab} \lt \frac{a+b}{2}
\end{equation*}
Faulty proof.
\begin{align*}
\sqrt{ab} \amp \lt \frac{a+b}{2}\\
ab \amp \lt \frac{(a+b)^2}{4}\\
4ab \amp \lt a^2+2ab+b^2\\
0 \amp \lt a^2-2ab+b^2\\
0 \amp \lt (a-b)^2
\end{align*}
Identify any issues with the proof as written above and give a correct proof.
18.
Let \(x,y\in\mathbb{R}\) such that \(x,y\geq0\text{.}\) Show that \(\sqrt{x+y}\leq \sqrt{x}+\sqrt{y}\text{.}\)
You may use the following without proof:
\begin{equation*}
\text{If } 0\leq a\leq b, \text{ then } \sqrt{a}\leq \sqrt{b}.
\end{equation*}
