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Module 5 · L1 of 20 ~30 min ⚡ +90 XP available

Introduction to Mathematical Proof

Can you ever be 100% certain a mathematical statement is true? Scientists rely on experiments, but mathematicians demand something stronger — a logical proof that leaves no room for doubt. In this lesson you'll learn what proof really is, the four main types used at HSC level, and what separates a convincing argument from an actual proof.

Today's hook — Is the statement "the sum of two even numbers is always even" true? Before reading on, try to write a convincing argument in two sentences. Is your argument truly a proof, or just a good example?

0/5QUESTS

Recall — your gut answer first

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Think about the statement: "The sum of two even numbers is always even." Without using algebra yet, write a short argument below explaining why you think this is (or isn't) true. Is checking a few examples enough?

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Think first — what counts as a proof?

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Here are two "arguments" that the sum of two even numbers is even:

Argument A (Examples)

$2 + 4 = 6$ (even), $8 + 10 = 18$ (even), $100 + 200 = 300$ (even). I've checked many cases, so it must always be true.

Argument B (Algebra)

Let $a = 2m$ and $b = 2n$. Then $a + b = 2m + 2n = 2(m+n)$, which is even for any integers $m, n$.

Only Argument B is a proof. Argument A, however convincing, only covers finitely many cases. A proof must work for all possible values — which requires algebra or logic, not examples.

What you'll master

Know

Key facts

A mathematical proof is a deductive argument establishing a statement for all cases
The four main proof types: direct, contradiction, induction, contrapositive
Vocabulary: axiom, theorem, lemma, corollary, conjecture

Understand

Concepts

Why checking examples is never sufficient as a proof
The difference between deductive and inductive reasoning
How each proof type is structured and when to use it

Can do

Skills

Write a complete direct proof with every step justified
Identify which proof type is most appropriate for a given statement
Distinguish valid proofs from flawed arguments

Key terms

Mathematical proofA logical deductive argument that establishes the truth of a statement beyond all doubt, for every case covered by the statement.

AxiomA self-evident truth accepted without proof; the starting point of a mathematical system (e.g. two points determine a unique line).

TheoremA mathematical statement that has been proved true using axioms and previously established results.

LemmaA minor result proved specifically to assist in proving a larger theorem.

CorollaryA result that follows so directly from a theorem that it requires little additional proof.

ConjectureA statement believed to be true based on evidence or intuition but not yet proved (e.g. Goldbach's Conjecture).

What is a mathematical proof?

core concept

A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt. Unlike scientific evidence — which is based on observation — a mathematical proof relies on deductive reasoning from axioms and previously established results.

Deductive vs inductive reasoning. In deductive reasoning you move from general rules to specific conclusions (this is what proofs use). In inductive reasoning you observe specific cases and conjecture a general rule. Note: "proof by mathematical induction" is actually a deductive method despite its name!

In the HSC, proof questions typically carry 3–5 marks and require clear, structured working. Every step must be justified — writing "obviously" or "clearly" is not enough.

Key rule: A proof is valid only if it works for every case in the domain of the statement. One counterexample is enough to disprove a statement.

A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt . Unlike scientific evidence — which is based on observation — a mathematical proof relies on deductive reasoning from axioms and previously established results.

Pause — copy the definition of mathematical proof and the key distinction between deductive and inductive reasoning into your book.

Quick check: A student tests the statement "all prime numbers are odd" by checking 3, 5, 7, 11, 13 and concludes it is true. What is wrong with this argument?

The four main types of proof

core concept

We just saw that a valid proof like "let $a=2m$, $b=2n$; then $a+b=2(m+n)$" works for all integers simultaneously, while checking examples only covers finitely many cases. That raises a question: which logical strategy should you choose for a given problem? This card answers it → by cataloguing the four proof types (direct, contradiction, induction, contrapositive) and their trigger situations.

At HSC level you need to know four proof strategies:

1. Direct Proof

Start from known facts and definitions. Use algebra/logic to reach the desired conclusion step by step. Best for statements about even/odd integers, divisibility, and algebraic identities.

2. Proof by Contradiction

Assume the opposite of what you want to prove. Show this assumption leads to a logical impossibility (contradiction). Conclude the original statement must be true.

3. Mathematical Induction

Prove a base case, then prove that truth for $n = k$ implies truth for $n = k+1$. Covers all positive integers like dominoes falling. This is the main focus of Module 5.

4. Proof by Contrapositive

Instead of proving $P \Rightarrow Q$, prove the logically equivalent statement $\neg Q \Rightarrow \neg P$. Sometimes much easier than a direct approach.

At HSC level you need to know four proof strategies:

Pause — copy all four proof strategies with a one-line trigger phrase for each (direct, contradiction, induction, contrapositive) into your book.

Did you get this? True or false: proving that $\neg Q \Rightarrow \neg P$ is logically equivalent to proving $P \Rightarrow Q$.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · DIRECT PROOF

Prove that the sum of two even integers is even.

Let $a$ and $b$ be even integers. Then $a = 2m$ and $b = 2n$ for some integers $m$ and $n$.

Translate "even" into algebra. This is the standard definition: an even integer equals twice some integer.

PROBLEM 2 · DIRECT PROOF

Prove that the product of two odd integers is odd.

Let $a$ and $b$ be odd integers. Write $a = 2m + 1$ and $b = 2n + 1$ for integers $m, n$.

Use the algebraic definition of an odd integer: it equals $2k + 1$ for some integer $k$.

PROBLEM 3 · CONTRAPOSITIVE

Prove: if $n^2$ is even, then $n$ is even.

We prove the contrapositive: if $n$ is odd, then $n^2$ is odd.

The contrapositive of "if $A$ then $B$" is "if not $B$ then not $A$." Here: if $n^2$ is even $\Rightarrow$ $n$ is even becomes if $n$ is odd $\Rightarrow$ $n^2$ is odd.

Fill the gap: In a direct proof that $n^2 + n$ is even, we write $n^2 + n = n(n+1)$. Since one of $n$ or $n+1$ is always , the product is even.

Misconceptions to fix · the 3 traps that cost marks

Trap 01

Checking examples is not a proof

Verifying a statement for 10, 100, or even 1 000 000 cases does not prove it for all cases. One counterexample disproves it, but no finite number of examples can prove it. Always use algebra or logic.

Trap 02

"Obvious" is not a justification

Every step in an HSC proof must be justified. Writing "it is clear that" or "obviously" loses marks. State the reason: "by the definition of even integers", "by factorising", etc.

Trap 03

Proving a specific case, not the general case

If asked to prove a statement for all integers $n$, writing "let $n = 3$" only proves it for that one value. Variables like $m$ and $n$ must represent arbitrary (general) integers.

Did you get this? True or false: to prove the statement "if $n$ is even then $n^2$ is even" by direct proof, you should let $n = 2m$ for an arbitrary integer $m$, then expand $n^2$.

Work mode · how are you completing this lesson?

Activities · practice with the ideas

Prove that the sum of an even integer and an odd integer is odd.

Prove that $n^2 + n$ is even for every integer $n$. (Hint: factorise.)

Give a counterexample to disprove: "the sum of two prime numbers is always odd."

State the contrapositive of: "If $x^2 > 4$ then $x > 2$."

Prove directly that the square of an even integer is divisible by 4.

Odd one out: Three of these are valid beginnings to a direct proof about integers. Which one is NOT a valid proof step?

Revisit your thinking

At the start of this lesson you wrote an argument for why the sum of two even numbers is always even. Now that you've seen a proper direct proof, look back at your argument.

A correct proof: let $a = 2m$, $b = 2n$. Then $a + b = 2(m+n)$, which is even since $m + n \in \mathbb{Z}$. $\blacksquare$

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Multiple choice

+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.

Short answer

ApplyBand 32 marks

Q1. Prove that the product of two even integers is divisible by 4. (2 marks)

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UnderstandBand 42 marks

Q2. Give an example of a conjecture that was believed true for a long time but was eventually disproved, OR explain why checking many cases cannot prove a statement true. (2 marks)

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AnalyseBand 53 marks

Q3. Prove by the contrapositive method that if $n^2$ is odd then $n$ is odd. (3 marks)

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Comprehensive answers (click to reveal)

Activity answers:

1. Let $a = 2m$, $b = 2n+1$. Then $a+b = 2m + 2n + 1 = 2(m+n) + 1$, which is odd. $\blacksquare$

2. $n^2 + n = n(n+1)$. One of $n, n+1$ is even, so their product has a factor of 2. Hence $n(n+1)$ is even. $\blacksquare$

3. $2 + 2 = 4$ (even), so the sum of two primes is not always odd (both can be 2).

4. Contrapositive: "If $x \leq 2$ then $x^2 \leq 4$." (Note: the original statement is also false for negative $x$, e.g. $x = -3$.)

5. Let $n = 2m$. Then $n^2 = 4m^2 = 4(m^2)$. Since $m^2 \in \mathbb{Z}$, $n^2$ is divisible by 4. $\blacksquare$

Q1 (2 marks): Let $a = 2m$, $b = 2n$. Then $ab = 4mn = 4(mn)$ [1]. Since $mn \in \mathbb{Z}$, $ab$ is divisible by 4 [1]. $\blacksquare$

Q2 (2 marks): Accept any of: Euler's conjecture (disproved 1966); Fermat's Last Theorem (conjecture for 350 years). For the explanation: a proof must cover all infinitely many cases simultaneously; checking examples only covers finitely many [2].

Q3 (3 marks): Contrapositive: if $n$ is even then $n^2$ is even [1]. Let $n = 2k$. Then $n^2 = 4k^2 = 2(2k^2)$, which is even [1]. Therefore the contrapositive is proved, and hence if $n^2$ is odd then $n$ is odd [1]. $\blacksquare$

Boss battle · The Proof Architect

earn bronze · silver · gold

Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

⚔ Enter the arena

Science Jump · platform challenge

Climb platforms by answering proof questions. Lighter alternative to the boss.

Mark lesson as complete

Tick when you've finished the practice and review.

← Y11 M4 L15 · Combinatorial Identities Lesson 2 · The Principle of Mathematical Induction →

Module overview · Extension 1 · Checkpoint 1