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

Work each one first, closed book. Reveal a hint only if you are stuck, the next step to see the first move, and the full working only to check.

Warm-up: remember

Write down the first five prime numbers.
Work out the value of $n^2 + 1$ when $n = 5$ .
Is $2$ an even number? Is $2$ a prime number?

Answers

$2$ , $3$ , $5$ , $7$ , $11$ .
$5^2 + 1 = 26$ .
Yes, and yes. It is the only even prime.

Practise with help

1. Which of these are statements? For each statement, say whether it is true or false. (a) $9$ is prime (b) Solve $x + 1 = 2$ (c) $4 + 5 = 9$ .

Hint

A statement is a sentence that is either true or false. Check each sentence for a question mark, an instruction, or a letter whose value is not fixed.

Next step

Take the first sentence on its own. "9 is prime" claims something specific, so test it by looking for a factor of $9$ .

Full working

(a) A statement, and false: $9 = 3 \times 3$ , so $9$ has a factor other than $1$ and itself.

(b) Not a statement: "Solve" is an instruction, not a claim.

2. Write the assumption and the conclusion of: "If a shape is a square, then it has four sides."

Hint

The assumption is the part after "if". The conclusion is the part after "then".

Next step

Write the sentence underneath in the form "if $A$ then $B$ ", then copy $A$ and $B$ into two separate lines.

Full working

Assumption: the shape is a square.

Conclusion: the shape has four sides.

3. Write the converse of the statement in question 2, and give a shape that shows the converse is false.

Hint

Swap the assumption and the conclusion. The converse starts with "If it has four sides".

Next step

Write the converse in full, then look for a four-sided shape that is not a square.

Full working

Converse: "If a shape has four sides, then it is a square."

The converse is false. A rectangle that is $3$ cm by $4$ cm has four sides, but it is not a square. The original statement is true and its converse is false, so the two directions are independent.

4. The statement is: "If $n$ is a multiple of $3$ , then $n$ is a multiple of $9$ ." Test it with $n = 3$ and $n = 9$ .

Hint

The assumption is "a multiple of $3$ ", so both values satisfy the assumption. Now check the conclusion for each value.

Next step

Work out $3 \div 9$ and $9 \div 9$ as whole-number divisions with remainders.

Full working

For $n = 3$ : the assumption holds, since $3 = 3 \times 1$ . The conclusion fails, since $3$ is not a multiple of $9$ .

For $n = 9$ : the assumption holds, and the conclusion holds, since $9 = 9 \times 1$ .

The value $n = 3$ shows the statement is false. One counter example is enough.

5. Disprove the statement "For every real number $x$ , $x^2 > x$ " by finding one value of $x$ .

Hint

Look for a value between $0$ and $1$ . Squaring a number in that range makes it smaller, not larger.

Next step

Try $x = \frac{1}{2}$ and work out $x^2$ .

Full working

Take $x = \frac{1}{2}$ . Then

x^2 = \frac{1}{4}.

Now

\frac{1}{4} < \frac{1}{2},

so $x^2 > x$ is false for this value. The statement says "for every real number", so one counter example disproves it.

6. Return to the chapter problem. For Claim B, work out the value of $n^2 + n + 41$ when $n = 40$ , and show that the value is not prime.

Hint

Substitute $40$ for $n$ and work out the three terms separately before adding them.

Next step

Work out $40^2$ , then add $40$ , then add $41$ , and look at the total.

Full working

For $n = 40$ ,

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

But

1681 = 41 \times 41,

so the value has a factor other than $1$ and itself. It is not prime, and Claim B is false.