Mathematics · Year 12 · Algebraic Proof and Functional Analysis · Autumn Term

Proof by Contradiction and Disproof

Students will learn to construct proofs by contradiction and effectively use counter-examples to disprove statements.

• National Curriculum Attainment Targets
• A-Level: Mathematics - Proof

About This Topic

Proof by contradiction requires students to assume the negation of a statement and derive a logical impossibility, such as proving √2 irrational. They suppose √2 = p/q in lowest terms, square both sides to find p even, set p = 2k, substitute to show q even, contradicting the lowest terms assumption. Disproof uses counterexamples, like rejecting 'every even number greater than 2 is the sum of two primes' with a case like 26, though Goldbach remains unproven; instead, simple claims like 'all primes greater than 2 are odd' wait no, correct ex: 'squares mod 4 never 3' counter with checks.

This topic anchors the Algebraic Proof unit in A-Level Mathematics, Autumn term, aligning with standards on constructing and critiquing proofs. Students design proofs for irrationals, analyze assumption flaws, and spot disproof errors, building precision for functional analysis.

Active learning suits this perfectly. When students role-play assumptions in pairs or collaboratively hunt counterexamples for posted statements, they internalize logic steps through trial and peer feedback, turning abstract reasoning into confident skill.

Key Questions

1. Design a proof by contradiction for the irrationality of a number.
2. Critique common errors in attempting to disprove a statement.
3. Analyze how the assumption of the opposite leads to a logical inconsistency in proof by contradiction.

Learning Objectives

• Design a proof by contradiction to demonstrate the irrationality of specific numbers, such as √3.
• Analyze the logical structure of a proof by contradiction, identifying the initial assumption and the derived contradiction.
• Critique common errors in disproof attempts, such as insufficient counterexamples or flawed reasoning.
• Evaluate the validity of mathematical statements by constructing counterexamples or by proving them true using contradiction.
• Explain the relationship between assuming the negation of a statement and arriving at a logical inconsistency.

Before You Start

Properties of Numbers

Why: Students need a solid understanding of number types (rational, irrational, prime, even, odd) to construct proofs and identify counterexamples.

Basic Algebraic Manipulation

Why: The ability to manipulate algebraic expressions is essential for constructing the logical steps in proofs, particularly when squaring equations or substituting values.

Key Vocabulary

Proof by Contradiction	A method of proof where one assumes the opposite of what is to be proven and shows that this assumption leads to a logical impossibility or contradiction.
Counterexample	A specific instance or case that shows a general statement or rule to be false.
Logical Impossibility	A statement or situation that cannot logically exist, often arising from contradictory conditions.
Assumption	A statement or proposition taken for granted, especially as a basis for argument or investigation, which is central to proof by contradiction.

Watch Out for These Misconceptions

Common Misconception

Any inconsistency disproves the assumption, even unrelated ones.

What to Teach Instead

True contradictions must follow logically from the assumption. Pair debates help students trace chains and spot irrelevancies, while group critiques refine valid paths.

Common Misconception

A single counterexample proves why a statement is false generally.

What to Teach Instead

Counterexamples show falsity but not full reason; they just violate universality. Collaborative hunts reveal patterns in counters, building deeper insight through shared examples.

Common Misconception

Proof by contradiction replaces direct proof always.

What to Teach Instead

It suits certain cases like irrationals; direct methods fit others. Relay activities let students compare methods actively, clarifying when each applies best.

Active Learning Ideas

Pairs: Assumption Debate

Assign statements like '√3 rational.' One student assumes true, other false; they build argument chains until contradiction emerges. Switch roles after 10 minutes, then pairs share strongest proofs with class.

30 min·Pairs

Small Groups: Counterexample Hunt

Provide 5 statements to disprove, such as 'n² + 1 always even.' Groups find and verify 2-3 counterexamples each, record with explanations, then rotate to critique others' work on posters.

40 min·Small Groups

Whole Class: Proof Relay

Project √2 irrational proof skeleton. Students line up; each adds one logical step verbally or on board, classmates vote yes/no with reasons before next turn, correcting as a group.

25 min·Whole Class

Individual: Custom Disproof

Students pick a false claim from handout, like 'all triangles equilateral,' craft counterexample with diagram and proof sketch, then pair-share for peer review.

20 min·Individual

Real-World Connections

• Computer scientists use proof by contradiction to verify the correctness of algorithms, ensuring that no unexpected or erroneous states can be reached during program execution, which is critical for secure software development.
• In cryptography, mathematicians employ proof by contradiction to demonstrate the security of encryption methods, proving that certain types of attacks are impossible under specific mathematical assumptions.

Assessment Ideas

Quick Check

Present students with the statement: 'All prime numbers are odd.' Ask them to write one sentence explaining why this statement is false and provide the specific counterexample. Then, ask them to explain what a proof by contradiction would look like for the statement 'All prime numbers are odd'.

Discussion Prompt

Pose the following: 'Imagine a student is trying to prove that the sum of two even numbers is always odd. They start by assuming the sum is odd and then show that this leads to the conclusion that an odd number must equal an even number. What is the flaw in their approach, and how would you correct it using proof by contradiction?'

Peer Assessment

In pairs, have students write a statement about numbers (e.g., 'The square of any rational number is rational'). One student writes a proof by contradiction, and the other writes a disproof by counterexample. They then swap and critique each other's work, checking for logical flow and accuracy of the contradiction or counterexample.

Frequently Asked Questions

What is proof by contradiction in A-Level Maths?

Proof by contradiction assumes the opposite of the claim and derives a contradiction, proving the original true. For √2 irrational, assume rational p/q lowest terms: leads to both even, impossible. Students practice by designing proofs and critiquing peers, aligning with UK A-Level proof standards for logical rigor.

How do you disprove a mathematical statement?

Use a counterexample: find one case violating the claim, like 3 disproves 'all primes even.' Verify it fits exactly. Class hunts and gallery walks ensure students explain why counters work, avoiding vague rejections and building disproof confidence.

Give an example of proof by contradiction for irrationality.

Prove √2 irrational: assume √2 = p/q, gcd(p,q)=1. Then p²=2q², p even so p=2k, 4k²=2q², q²=2k², q even: contradicts gcd=1. Relay activities make students own each step collaboratively.

How does active learning help with proof by contradiction?

Active methods like pair debates on assumptions or group counterexample challenges make logic interactive. Students spot flaws in real time through peer feedback, internalize steps via relays, and gain confidence constructing proofs, far beyond passive notes for A-Level success.