6.5: Exercise

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$Hammack.jpg$
Contributed by Richard Hammack
Professor (Mathematics) at Virginia Commonwealth University

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A. Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.)

Exercise $\PageIndex{1}$

Suppose $n \in \mathbb{Z}$. If n is odd, then $n^2$ is odd.

Exercise $\PageIndex{2}$

Suppose $n \in \mathbb{Z}$. If $n^2$ is odd, then n is odd.

Exercise $\PageIndex{3}$

Prove that $\sqrt[3]{2}$ is irrational.

Exercise $\PageIndex{4}$

Prove that $\sqrt{6}$ is irrational.

Exercise $\PageIndex{5}$

Prove that $\sqrt{3}$ is irrational.

Exercise $\PageIndex{6}$

If $a, b \in \mathbb{Z}$, then $a^2-4b-2 \ne 0$.

Exercise $\PageIndex{7}$

If $a, b \in \mathbb{Z}$, then $a^2-4b-3 \ne 0$.

Exercise $\PageIndex{8}$

Suppose $a, b, c \in \mathbb{Z}$. If $a^2+b^2 = c^2$, then a or b is even.

Exercise $\PageIndex{9}$

Suppose $a, b \in \mathbb{R}$. If a is rational and ab is irrational, then b is irrational.

Exercise $\PageIndex{10}$

There exist no integers a and b for which $21a + 30b = 1$.

Exercise $\PageIndex{11}$

There exist no integers a and b for which $18a + 6b = 1$.

Exercise $\PageIndex{12}$

For every positive $x \in \mathbb{Q}$, there is a positive $y \in \mathbb{Q}$ for which $y < x$.

Exercise $\PageIndex{13}$

For every $x \in [\pi/2, \pi]$, $sinx-cosx \ge 1$.

Exercise $\PageIndex{14}$

If A and B are sets, then $A \cap (B-A)= \emptyset$.

Exercise $\PageIndex{15}$

If $b \in \mathbb{Z}$ and $b|k$ for every $k \in \mathbb{N}$, then $b = 0$.

Exercise $\PageIndex{16}$

If a and b are positive real numbers, then $a+b \ge 2\sqrt{ab}$.

Exercise $\PageIndex{17}$

For every $n \in \mathbb{Z}$, $4 \nmid (n^2+2)$.

Exercise $\PageIndex{18}$

Suppose $a, b \in \mathbb{Z}$. If $4|(a^2+b^2)$, then a and b are not both odd.

B. Prove the following statements using any method from Chapters 4, 5 or 6.

Exercise $\PageIndex{19}$

The product of any five consecutive integers is divisible by 120. (For example, the product of 3,4,5,6 and 7 is 2520, and $2520 = 120 \cdot 21$.)

Exercise $\PageIndex{20}$

We say that a point P = (x, y) in $\mathbb{R}^2$ is rational if both x and y are rational. More precisely, P is rational if $P =(x,y) \in \mathbb{Q}^2$. An equation $F(x,y)=0$ is said to have a rational point if there exists $x_{0}, y_{0} \in \mathbb{Q}$ such that $F(x_{0}, y_{0}) = 0$. For example, the curve $x^2+y^2-1 = 0$ has rational point $(x_{0}, y_{0}) = (1, 0)$. Show that the curve $x^2+y^2-3 = 0$ has no rational points.

Exercise $\PageIndex{21}$

Exercise 20 (above) involved showing that there are no rational points on the curve $x^2+y^2-3=0$. Use this fact to show that $\sqrt{3}$ is irrational.

Exercise $\PageIndex{22}$

Explain why $x^2+y^2-3 = 0$ not having any rational solutions (Exercise 20) implies $x^2+y^2-3k = 0$ has no rational solutions for k an odd, positive integer.

Exercise $\PageIndex{23}$

Use the above result to prove that $\sqrt{3^k}$ is irrational for all odd, positive k.

Exercise $\PageIndex{24}$

The number $log_{2} 3$ is irrational.