Direct proof and contradiction | Year 11

01 / LEARN

The idea, step by step

Direct proof

A direct proof starts from the hypotheses and follows logically justified steps to the conclusion. Use definitions: an even integer has the form $2 k$ ; an odd integer has the form $2 k + 1$ ; divisibility means an integer multiple.

Use different letters for unrelated integers. Writing two even integers as $2 k$ and $2 k$ accidentally assumes they are equal. Start with $2 a$ and $2 b$ , then show the result has the required form.

Proof by contradiction

For contradiction, assume the hypotheses and the negation of the desired conclusion. Derive something impossible, such as an integer being both even and odd. State exactly where the contradiction arises.

Do not merely assume the conclusion and manipulate it until it looks plausible. Each implication must follow from your assumptions. When a proof uses lowest terms, explain why a common factor contradicts that choice.

A common trap

Beginning a proof with a single convenient value, such as $n = 3$ .

Instead: Use an arbitrary permitted value to establish a universal conclusion.

02 / WORKED EXAMPLES

See the thinking

Example 1

The sum of two odd integers

Prove that the sum of two odd integers is even.

Let the integers be $2 a + 1$ and $2 b + 1$ , with $a, b \in ℤ$ .
Their sum is $2 a + 2 b + 2 = 2 (a + b + 1)$ .
Since $a + b + 1$ is an integer, the sum is even.

The sum is even.

Example 2

No smallest positive real number

Prove that there is no smallest positive real number.

Suppose a smallest positive real number $s$ exists.
Then $s / 2$ is positive, and $s / 2 < s$ .
This contradicts the assumption that $s$ is the smallest.

No smallest positive real number exists.

Check your understanding

Prove that the square of an even integer is even.

Show answer

$n = 2 k \Rightarrow n^{2} = 2 (2 k^{2})$ .

The expression is twice an integer, which is the definition of even.

03 / WORKSHEET

Your turn

6 questions · 16 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Prove that the sum of two even integers is even.
2 marks
Prove that the product of two odd integers is odd.
3 marks
Prove that consecutive integers have an even product.
3 marks
Prove by contradiction that a real number cannot be both positive and negative.
2 marks
Prove by contradiction that there is no largest integer.
3 marks
A proof writes two odd numbers as $2 k + 1$ and $2 k + 1$ . Explain the gap and repair it.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Prove that the sum of two even integers is even.
2 marks
Write $2 a + 2 b = 2 (a + b)$ , where $a + b$ is an integer.
It is even.
Prove that the product of two odd integers is odd.
3 marks
$(2 a + 1) (2 b + 1) = 2 (2 a b + a + b) + 1$ . The bracket is an integer.
It is odd.
Prove that consecutive integers have an even product.
3 marks
One of $n, n + 1$ is even, so their product contains a factor of 2.
$n (n + 1)$ is even.
Prove by contradiction that a real number cannot be both positive and negative.
2 marks
Assume $x > 0$ and $x < 0$ . These require $x$ to lie on opposite sides of zero simultaneously, contradicting the order properties of real numbers.
Impossible.
Prove by contradiction that there is no largest integer.
3 marks
Suppose $N$ is the largest integer. Then $N + 1$ is an integer larger than $N$ , a contradiction.
No largest integer exists.
A proof writes two odd numbers as $2 k + 1$ and $2 k + 1$ . Explain the gap and repair it.
3 marks
Use $2 a + 1$ and $2 b + 1$ with independent integer parameters; the argument then applies to every pair.
It assumes the odd numbers are equal.

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Rational/irrational numbers and simple number proofs

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: geometric reasoning and real-number/proof material, regrouped into a school chapter; all exercises are original.

Official curriculum source ↗

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