Direct proof and contradiction | Year 11

01 / LEARN

Understand the idea.

Direct proof

A direct proof begins with the hypotheses and rewrites them using definitions. To prove a result is even, aim to express it as twice an integer. To prove it is odd, aim for one more than twice an integer.

Choose independent letters for unrelated numbers. Two arbitrary odd integers can be $2 a + 1$ and $2 b + 1$ , with $a, b \in ℤ$ . Using the same parameter for both would restrict the argument to equal numbers.

Proof by contradiction

In proof by contradiction, assume the hypotheses together with the negation of the desired conclusion. Follow valid deductions until they conflict with an assumption or an established fact. Name that conflict explicitly before rejecting the contradictory assumption.

An existence claim often suggests a construction: if a smallest positive number existed, could you build a smaller positive one? The construction must remain in the same domain; a smaller number outside it would produce no contradiction.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

The sum of two odd integers

Prove that the sum of two odd integers is even.

Worked solution

1
Represent any two odd integers independently.
$u = 2 a + 1, v = 2 b + 1, a, b \in ℤ$
The definition of odd applies to each integer without assuming the two are equal.
2
Add and factor out two.
$\begin{aligned} u + v & = 2 a + 2 b + 2 \\ = 2 (a + b + 1) \end{aligned}$
Factoring exposes the form required for an even integer.
3
Identify the integer factor.
$a + b + 1 \in ℤ$
Integers are closed under addition, so the sum of the two odd integers is even.

Answer

The sum is even.

EXAMPLE 02

No smallest positive real number

Prove that there is no smallest positive real number.

Worked solution

1
Assume the opposite of the claim.
$s > 0$
Suppose $s$ is a smallest positive real number; every positive real number would then be at least $s$ .
2
Construct a smaller permitted number.
$0 < \frac{s}{2} < s$
Dividing a positive real number by two preserves positivity and strictly decreases it.
3
Identify the contradiction.
The positive real number $s / 2$ is smaller than the assumed smallest one. Therefore no smallest positive real number exists.

Answer

No smallest positive real number exists.

03 / YOUR TURN

Try it yourself.

Check your understanding

Prove that the square of an even integer is even.

Show answer & reasoning

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

For any even integer, write $n = 2 k$ with $k \in ℤ$ . Then $n^{2} = 4 k^{2} = 2 (2 k^{2})$ , and $2 k^{2}$ is an integer, so the square is even.

Keep practising

6 worksheet questions · 16 marks

Open worksheet 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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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 ↗

In this lesson

Understand the idea.

Direct proof

Proof by contradiction

See the method in action.

The sum of two odd integers

Represent any two odd integers independently.

Add and factor out two.

Identify the integer factor.

No smallest positive real number

Assume the opposite of the claim.

Construct a smaller permitted number.

Identify the contradiction.

Try it yourself.

Check your understanding

Keep practising