
6.5: Exercise


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.