Chapter 1: Review

The chapter problem, solved

Claim A. For every whole number $n$ , the value of $n^2 + n$ is even.

Factorise the expression:

n^2 + n = n(n + 1).

The numbers $n$ and $n + 1$ are consecutive, so one of them is even. The product of a whole number and an even number is even, so $n(n + 1)$ is even. Claim A is true.

Claim B. For every whole number $n$ , the value of $n^2 + n + 41$ is prime.

Take $n = 40$ . Then

n^2 + n + 41 = 1600 + 40 + 41 = 1681.

But

1681 = 41 \times 41,

so the value is not prime. A single counter example disproves the claim. Claim B is false. The formula works for the values $n = 0$ to $39$ , and that long run of successes is exactly the trap: evidence is not proof.

Claim C. There is a largest prime number.

Assume, for contradiction, that the primes can all be listed:

p_1, p_2, p_3, \ldots, p_n.

Form $N = p_1 p_2 \cdots p_n + 1$ . Each prime $p_i$ divides the product, so $N$ leaves remainder $1$ when divided by each $p_i$ . No prime in the list divides $N$ . But $N > 1$ , so $N$ has a prime factor, and that factor must be in the complete list of primes. This is a contradiction. Claim C is false: there is no largest prime.

Three claims, three different methods: deduction proved a "for all" statement, one counter example disproved another, and contradiction settled the third.

Mixed practice — choose the method first

For each question, say which method you will use — deduction, exhaustion, counter example, or contradiction — and why. Only then calculate.

Prove that the sum of two consecutive odd numbers is a multiple of $4$ .
Show that $137$ is prime.
Disprove the claim "Every whole number of the form $n^2 + 1$ is prime."
Prove that $\sqrt{5}$ is irrational.
Show that the sum of three consecutive whole numbers is a multiple of $3$ .
Disprove the claim "For all real numbers $x$ and $y$ , if $x > y$ then $x^2 > y^2$ ."
Prove that there is no largest odd number.
Show that every even number from $4$ to $14$ is the sum of two prime numbers.
Prove that $n^2 - n$ is even for every whole number $n$ .
Prove that the product of two rational numbers is rational.

Answers

Deduction. Consecutive odd numbers are $2n + 1$ and $2n + 3$ . Their sum is $4n + 4 = 4(n + 1)$ , a multiple of $4$ .
Exhaustion. Since $11^2 = 121 \leq 137$ and $13^2 = 169 > 137$ , test the primes $2$ , $3$ , $5$ , $7$ and $11$ . The remainders are $1$ , $2$ , $2$ , $4$ and $5$ , so none divides $137$ . Therefore $137$ is prime.
Counter example. Take $n = 7$ . Then $n^2 + 1 = 50$ , and $50 = 2 \times 25$ , so the value is not prime.
Contradiction. Assume $\sqrt{5} = a/b$ in lowest terms. Then $a^2 = 5b^2$ , so $5$ divides $a^2$ and therefore $5$ divides $a$ . Write $a = 5k$ . Then $25k^2 = 5b^2$ , so $5k^2 = b^2$ , and $5$ divides $b$ . Both $a$ and $b$ are multiples of $5$ , contradicting lowest terms. Therefore $\sqrt{5}$ is irrational.
Deduction. The numbers are $n$ , $n + 1$ and $n + 2$ . Their sum is $3n + 3 = 3(n + 1)$ , a multiple of $3$ .
Counter example. Take $x = 1$ and $y = -2$ . Then $x > y$ , but $x^2 = 1 < 4 = y^2$ .
Contradiction. Assume $M$ is the largest odd number, so $M = 2k + 1$ . Then $M + 2 = 2k + 3 = 2(k + 1) + 1$ is odd and greater than $M$ . This contradicts the choice of $M$ , so there is no largest odd number.
Exhaustion. $4 = 2 + 2$ , $6 = 3 + 3$ , $8 = 3 + 5$ , $10 = 3 + 7$ , $12 = 5 + 7$ and $14 = 3 + 11$ . The list covers every even number from $4$ to $14$ .
Deduction. Factorise: $n^2 - n = n(n - 1)$ . The numbers $n - 1$ and $n$ are consecutive, so one is even, and the product is even.
Deduction. Let the rational numbers be $\frac{a}{b}$ and $\frac{c}{d}$ , with $b \neq 0$ and $d \neq 0$ . Their product is $\frac{ac}{bd}$ , and $bd \neq 0$ , so the product is a fraction of whole numbers. Therefore it is rational.

What you can now do

read a mathematical statement and name its assumption and conclusion;
tell "for all" from "there is", and use "converse", "necessary" and "sufficient";
prove a general statement by deduction, using the general forms $2n$ and $2n + 1$ ;
prove a statement about a finite set by exhaustion, and say when a list of cases is complete;
disprove a "for all" statement with one counter example;
prove a statement by contradiction, including the irrationality of $\sqrt{2}$ and the infinity of the primes;
choose a method for a new claim and say why it fits.

Your relevance reflection

Return to the prompt in the chapter opener: a rule that seemed true in every example you tried, but turned out to have an exception. Write one more example now that you have finished the chapter. Has the chapter changed what you notice about a claim such as "this always works"? Write a sentence saying which kind of evidence would make you trust such a claim, and which kind would make you doubt it.