1.4 Disproof by counter example — Practise

Work through these on your own. A disproof needs two things: a value for which the assumption holds, and a clear reason why the conclusion fails. Write both. The last item asks you to choose the method: say which one fits and why before you start.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

The claim is "For every whole number $n$ , the value of $n^2 + n + 41$ is prime." Find a counter example.
The claim is "Every prime number is odd." Find a counter example.
The claim is "If $n$ is prime, then $2^n - 1$ is prime." Find a counter example.
The claim is "For every real number $x$ , $x^2 > x$ ." Find a counter example.
The claim is "For all real numbers $x$ and $y$ , if $x^2 > y^2$ then $x > y$ ." Find a counter example.
The claim is "If a whole number is a multiple of $6$ , then it is a multiple of $12$ ." Find a counter example.
The claim is "For every whole number $n$ , the value of $n^2 - n + 11$ is prime." Find a counter example.
The claim is "The sum of two irrational numbers is irrational." Find a counter example.
The claim is "The sum of two prime numbers is even." Find a counter example.
The claim is "Every triangle whose sides are whole numbers has a right angle." Find a counter example.
Choose the method for each part before you answer, then explain your choice. (a) Prove that the sum of two odd numbers is even. (b) Decide whether every whole number of the form $n^2 + 1$ is prime.

Answers and concise solutions

1. The counter example

Answer: Take $n = 40$ . Then $n^2 + n + 41 = 1600 + 40 + 41 = 1681 = 41 \times 41$ . The value is not prime, so the claim is false.

2. The counter example

Answer: The number $2$ is prime, because its only factors are $1$ and $2$ . It is even, not odd. So the claim is false.

3. The counter example

Answer: Take $n = 11$ , which is prime. Then $2^{11} - 1 = 2048 - 1 = 2047$ , and $2047 = 23 \times 89$ . A product of two numbers greater than $1$ is not prime, so the claim is false.

4. The counter example

Answer: Take $x = \frac{1}{2}$ . Then $x^2 = \frac{1}{4}$ , and $\frac{1}{4} < \frac{1}{2}$ . So $x^2 > x$ fails and the claim is false.

5. The counter example

Answer: Take $x = -3$ and $y = 2$ . Then $x^2 = 9 > 4 = y^2$ , but $x = -3 < 2 = y$ . The claim is false, because squaring hides the sign.

6. The counter example

Answer: Take the number $6$ . It is a multiple of $6$ , but $6$ is not a multiple of $12$ : it leaves remainder $6$ . The claim is false.

7. The counter example

Answer: Take $n = 11$ . Then $n^2 - n + 11 = 121 - 11 + 11 = 121 = 11 \times 11$ . The value is not prime, so the claim is false.

8. The counter example

Answer: Take $\sqrt{2}$ and $-\sqrt{2}$ . Both are irrational, but their sum is $\sqrt{2} - \sqrt{2} = 0$ , and $0$ is rational. The claim is false.

9. The counter example

Answer: Take the primes $2$ and $3$ . Their sum is $2 + 3 = 5$ , which is odd. The claim fails because one of the primes can be $2$ , the only even prime.

10. The counter example

Answer: Take the triangle with sides $2$ , $3$ and $4$ . For a right angle the longest side would need $4^2 = 2^2 + 3^2$ . But $16$ is not $13$ , so there is no right angle. The claim is false.

11. Which method, and why

Answer: (a) Deduction. The statement is "for all" odd numbers, so one example proves nothing. Write them as $2m + 1$ and $2n + 1$ ; then $(2m + 1) + (2n + 1) = 2(m + n + 1)$ , a multiple of $2$ . (b) Counter example. The claim is "for all", so one failing value disproves it. Take $n = 7$ : $n^2 + 1 = 50 = 2 \times 25$ , which is not prime. The method follows from what is being asked: to prove a "for all" statement use deduction, to disprove one look for a counter example.