1.1 Statements, assumptions and the structure of a proof — Practise

Work through these on your own. For each item, write down whether you are proving a "for all" statement, disproving it, or showing "there is". The method matters as much as the answer.

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

Decide whether each statement is true or false. (a) $2 + 3 = 6$ (b) $12$ is a multiple of $4$ (c) $7 < 3$ .
A statement says: "If a whole number is a multiple of $6$ , then it is a multiple of $3$ ." (a) Write the assumption. (b) Write the conclusion. (c) Test the statement with the number $42$ .
A statement says: "If $n$ is a multiple of $4$ , then $n$ is even." (a) Write the converse. (b) Give a value of $n$ that shows the converse is false.
The statement "For every real number $x$ , $x^2 \geq x$ " is false. Find a value of $x$ that shows this.
Show that there is a whole number $n$ for which $n^2 + n + 1$ is a multiple of $3$ .
Work out the value of $n^2 + n + 41$ for $n = 1$ , $n = 2$ and $n = 3$ .
Decide whether each number is prime: (a) $9$ (b) $11$ .
A student checks $n = 1, 2, 3, 4, 5$ and finds that $n^2 + n + 41$ is prime each time. The student says: "This proves the claim for every whole number $n$ ." Explain what is wrong with the reasoning.
A whole number $n$ is a multiple of $6$ . Show that $n$ is also a multiple of $3$ .
The statement "If $n$ is a multiple of $3$ , then $n$ is a multiple of $9$ " is false. Give a value of $n$ that shows this.

Answers and concise solutions

1. True or false

Answer: (a) false, because $2 + 3 = 5$ (b) true, because $12 = 3 \times 4$ (c) false, because $7 > 3$ .

2. Assumption, conclusion and a test

Answer: Assumption: the number is a multiple of $6$ . Conclusion: the number is a multiple of $3$ . For $42$ : $42 = 6 \times 7$ , so the assumption holds, and $42 = 3 \times 14$ , so the conclusion holds too.

3. The converse and a counter example

Answer: (a) Converse: "If $n$ is even, then $n$ is a multiple of $4$ ." (b) Take $n = 2$ . The number $2$ is even, but $2$ leaves remainder $2$ on division by $4$ , so it is not a multiple of $4$ .

4. A value of $x$

Answer: Take $x = \frac{1}{2}$ . Then $x^2 = \frac{1}{4}$ , and $\frac{1}{4} < \frac{1}{2}$ , so $x^2 \geq x$ fails.

5. A value of $n$

Answer: For $n = 1$ , $n^2 + n + 1 = 3$ , which is a multiple of $3$ . One example is enough, because the statement says "there is".

6. The three values

Answer: (a) $43$ (b) $47$ (c) $53$ .

7. Prime or composite

Answer: (a) $9 = 3 \times 3$ , so $9$ is not prime. (b) $\sqrt{11}$ lies between $3$ and $5$ , so the only primes to test are $2$ and $3$ . Neither divides $11$ , so $11$ is prime.

8. The flaw in the reasoning

Answer: The five values are prime, so the checking is correct. But the claim is about every whole number, and there are infinitely many. A finite check is evidence, not proof: an untested value might still fail, and one later does. A proof needs a general argument that covers every value at once.

9. Show the conclusion

Answer: Write $n = 6k$ , because $n$ is a multiple of $6$ . Then $n = 3 \times (2k)$ . The factor $2k$ is a whole number, so $n$ is a multiple of $3$ .

10. A value of $n$

Answer: Take $n = 3$ . The number $3$ is a multiple of $3$ , but it is not a multiple of $9$ . One such value is enough to show the statement is false.