Chapter 1: Proof and mathematical argument

The chapter problem: which claims can we trust?

Three claims are on the table.

• Claim A. For every whole number $n$, the value of $n^2 + n$ is even.
• Claim B. For every whole number $n$, the value of $n^2 + n + 41$ is prime.
• Claim C. There is a largest prime number.

By the end of the chapter you will decide whether each claim is true or false, and you will prove your decision. A guess is not a proof: each answer needs an argument that covers every case at once.

First contact — no proof yet. Work out $n^2 + n + 41$ for $n = 1$, $n = 2$ and $n = 3$. All three values are prime. Would you bet that Claim B is true? Write down one sentence saying what would make you doubt it.

By the end of this chapter you will be able to

• say what a statement, an assumption and a conclusion are;
• tell the difference between "for all" and "there is";
• prove a general statement by deduction;
• prove a statement about a finite list of cases by exhaustion;
• disprove a statement by finding one counter example;
• prove a statement by contradiction, including the fact that $\sqrt{2}$ is irrational.

Where it is used

Proof is the reason mathematics works. A calculation answers one question; a proof answers every version of it at once. Every later chapter in this book leans on the ideas here: when you show that a formula holds for all $x$, or that a result cannot happen, you are writing a proof. At A level, argument and proof carry a quarter of the marks, across all three papers.

Your turn to connect it

Think of a rule that seemed true in every example you tried, but turned out to have an exception. It could be a rule of thumb about money, travel, games or sport. Write two examples: one from outside mathematics, and one from mathematics.

Roadmap

1. 1.1 Statements, assumptions and the structure of a proof
2. 1.2 Proof by deduction
3. 1.3 Proof by exhaustion
4. 1.4 Disproof by counter example
5. 1.5 Proof by contradiction
6. Review