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    Proof — WJEC A-Level Mathematics

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    Proof explained

    This topic covers the fundamental structure of mathematical proof, requiring students to proceed from given assumptions through logical steps to a conclusion.

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    It specifically includes proof by deduction, proof by exhaustion, and disproof by counter-example, with a specific requirement to prove the laws of logarithms.

    What to demonstrate

    1. Clear logical progression from assumptions to conclusion
    2. Correct use of mathematical notation and connecting language
    3. Accurate application of proof by deduction for laws of logarithms
    Show all 5 objectives
    1. Correct identification of counter-examples for disproof
    2. Systematic coverage of all cases in proof by exhaustion

    Proof exam tips

    Topic Overview

    Proof is the foundation of all mathematics. In the WJEC A-Level specification, proof is not just a topic but a skill that permeates every area of the course. You will learn to construct logical arguments to establish the truth of mathematical statements, using a variety of methods. This topic is assessed both in pure mathematics and in the context of mechanics and statistics, where you may need to prove results or justify steps in your working.

    The key methods of proof you need to master are: direct proof, proof by contradiction, proof by exhaustion, proof by counterexample, and proof by induction (for sequences and series). Each method has its own structure and typical applications. For example, direct proof is used to show that if a condition holds, then a conclusion follows, while proof by contradiction is powerful for statements that are difficult to prove directly, such as the irrationality of √2.

    Understanding proof is crucial for developing mathematical reasoning and for achieving the highest grades. In exams, proof questions often require you to show a given result, and marks are awarded for clear logical steps and correct notation. You must be able to choose the appropriate method and present your argument in a structured way. Mastery of proof also helps in other topics like algebra, trigonometry, and calculus, where you may need to justify identities or derive formulas.

    Key Concepts
    • →Direct proof: Start from known facts or assumptions and use logical steps to reach the conclusion. For example, proving that the sum of two even numbers is even.
    • →Proof by contradiction: Assume the opposite of what you want to prove, then show that this leads to a contradiction. Classic example: proving √2 is irrational.
    • →Proof by exhaustion: Check all possible cases individually. This is used when the statement can be split into a finite number of cases, e.g., proving that all integers from 1 to 10 satisfy a property.
    • →Proof by counterexample: To disprove a statement, find a single example where it fails. For instance, to disprove 'all prime numbers are odd', use the counterexample 2.
    • →Proof by induction: Used for statements involving natural numbers. Show the base case (e.g., n=1) is true, then assume true for n=k and prove it for n=k+1. Common in sequences and divisibility.
    Marking Points
    • Clear logical progression from assumptions to conclusion
    • Correct use of mathematical notation and connecting language
    • Accurate application of proof by deduction for laws of logarithms
    • Correct identification of counter-examples for disproof
    • Systematic coverage of all cases in proof by exhaustion
    Examiner Tips
    • 💡Ensure every step of a deductive proof is justified by a previous statement or definition
    • 💡When using disproof by counter-example, a single specific case that contradicts the statement is sufficient
    • 💡For proof by exhaustion, ensure you have explicitly listed all possible cases and verified each one
    • 💡Practice the formal proof of the laws of logarithms as this is explicitly mentioned in the guidance
    • 💡Use precise mathematical language; avoid vague or colloquial explanations
    • 💡Always state the method you are using at the start of your proof. For example, 'We will prove this by contradiction.' This helps the examiner follow your reasoning and ensures you get method marks.
    • 💡In proof by induction, clearly label the base case, the inductive hypothesis, and the inductive step. Use the phrase 'Assume true for n=k' and then show it leads to 'true for n=k+1'. Do not skip steps.
    • 💡When disproving a statement, a single counterexample is sufficient. Make sure your counterexample is clearly presented and that you explain why it disproves the statement. Avoid vague statements like 'it doesn't work for some numbers'.
    Common Mistakes
    • Failing to state assumptions clearly at the start of a proof
    • Using examples to 'prove' a general statement instead of using algebraic deduction
    • Incomplete exhaustion in proof by exhaustion (missing cases)
    • Incorrect use of logical symbols or connecting language
    • Assuming the result to be proved as part of the working
    • Misconception: 'Proof by contradiction is the same as proving the contrapositive.' Correction: They are different. In proof by contradiction, you assume the statement is false and derive a contradiction. In proving the contrapositive, you prove 'if not Q then not P' instead of 'if P then Q'.
    • Misconception: 'Proof by exhaustion means checking a few examples.' Correction: Exhaustion requires checking every possible case. If there are infinitely many cases, exhaustion is not possible; you need another method.
    • Misconception: 'In proof by induction, you only need to prove the inductive step.' Correction: You must also prove the base case. Without the base case, the induction is invalid.
    Frequently Asked Questions
    How do I know which proof method to use?
    Look at the statement you need to prove. If it's a universal statement (e.g., 'all even numbers...'), direct proof or exhaustion may work. If it's a negative statement (e.g., 'no integer satisfies...'), contradiction is often best. For statements about natural numbers, induction is common. Practice identifying the structure: 'if P then Q' often suggests direct proof, while 'P is false' suggests contradiction. Also, consider the number of cases: if finite, exhaustion; if infinite, another method.
    Can I use proof by exhaustion for infinite sets?
    No, proof by exhaustion only works when there are a finite number of cases to check. For infinite sets, you need a different method like direct proof or induction. However, sometimes you can split an infinite set into a finite number of categories (e.g., even and odd numbers) and prove each category separately, which is still a form of exhaustion if the categories are finite.
    What is the difference between proof by contradiction and proof by contrapositive?
    Proof by contradiction assumes the statement is false and derives a contradiction, often without a specific target. Proof by contrapositive proves 'if not Q then not P' instead of 'if P then Q'. They are related but not the same. For example, to prove 'if n^2 is even, then n is even', you could use contrapositive: 'if n is odd, then n^2 is odd'. Alternatively, you could assume n^2 even and n odd and derive a contradiction. Both are valid, but contrapositive is more direct in some cases.
    How do I write a proof by induction for divisibility?
    First, state the proposition P(n): 'n^3 - n is divisible by 3'. Base case: show P(1) is true (1-1=0, divisible by 3). Inductive hypothesis: assume P(k) is true, i.e., k^3 - k = 3m for some integer m. Inductive step: consider (k+1)^3 - (k+1) = k^3 + 3k^2 + 3k + 1 - k - 1 = (k^3 - k) + 3(k^2 + k). By hypothesis, k^3 - k = 3m, so the expression becomes 3m + 3(k^2 + k) = 3(m + k^2 + k), which is divisible by 3. Hence P(k+1) is true. Conclude by induction that P(n) holds for all natural numbers n.
    What is a counterexample and how do I find one?
    A counterexample is a specific case that shows a general statement is false. To find one, think of simple numbers or cases that might break the rule. For example, to disprove 'all prime numbers are odd', the counterexample is 2, which is prime and even. For algebraic statements, try small integers or fractions. If the statement involves variables, try to find values that make the left side different from the right side. Always check your counterexample carefully to ensure it truly violates the statement.
    Do I need to memorise all the proof methods?
    Yes, you should be familiar with all the methods listed in the specification: direct proof, contradiction, exhaustion, counterexample, and induction. However, you don't need to memorise every possible proof; instead, understand the structure of each method and practice applying them to different types of statements. In exams, you will be expected to choose the appropriate method and execute it correctly. Regular practice with past paper questions will help you become confident.

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