A-Level Proof by contradiction Practice Questions

Free A-Level Proof by contradiction practice questions with full step-by-step worked solutions. Covers primes, number-theory, rationals, contradiction. Practise exam-style problems and check your method.

primesnumber-theoryrationalscontradictionnegationirrational
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Find the smallest prime number greater than 77.
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Worked solution

  1. Interpret the task

    find the smallest prime>7\text{find the smallest prime}>7

    We test the integers above 7 for primality.

  2. Identify the first prime

    11 is prime11\ \text{is prime}

    This is the first integer above 7 with no proper divisors.

  3. State the answer

    1111

    This is the required prime.

Answer
1111
Question 2
2 markseasy
Assuming qq is the smallest positive rational, which value gives the contradiction?
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Worked solution

  1. State the claim

    there is no smallest positive rational\text{there is no smallest positive rational}

    We show no positive rational can be least.

  2. This contradicts q being smallest

    contradiction\text{contradiction}

    q was assumed to be the least positive rational.

  3. Conclude the claim

    no smallest positive rational exists\text{no smallest positive rational exists}

    The assumption of a least value is impossible.

Answer
q2\frac{q}{2}, since it is positive, rational and smaller than qq
Question 3
3 marksintermediate
Which is the correct first step of a proof by contradiction that log⁡23\log_2 3 is irrational?
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Worked solution

  1. State the claim

    log⁡23 is irrational\log_{2}3\ \text{is irrational}

    We show it is not a ratio of integers.

  2. Assume the opposite

    log⁡23=pq\log_{2}3=\frac{p}{q}

    Suppose it equals a fraction with positive integers p and q.

  3. Rewrite in exponential form

    2p/q=32^{p/q}=3

    Undo the logarithm.

  4. Raise both sides to the power q

    2p=3q2^{p}=3^{q}

    Remove the fractional exponent.

  5. Compare parity of the two sides

    2p is even, 3q is odd2^{p}\ \text{is even},\ 3^{q}\ \text{is odd}

    An even number cannot equal an odd number.

  6. Conclude the claim

    log⁡23 is irrational\log_{2}3\ \text{is irrational}

    The parity contradiction refutes the assumption.

Answer
Assume log⁡23=pq\log_2 3=\frac{p}{q} with positive integers p,qp,q
Question 4
5 markshard
Which of the following correctly describes proof by contradiction (as opposed to a direct proof)?
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Worked solution

  1. Recall a direct proof

    assume P, deduce Q\text{assume}\ P,\ \text{deduce}\ Q

    A direct proof reasons forward from the hypothesis.

  2. Recall a proof by contradiction

    assume ¬Q, derive P∧¬P\text{assume}\ \lnot Q,\ \text{derive}\ P\wedge\lnot P

    A contradiction proof assumes the negation and reaches an impossibility.

  3. Compare the two approaches

    contradiction targets an impossibility\text{contradiction targets an impossibility}

    The defining feature is the impossibility that is reached.

  4. Work only with consequences of the assumption

    each line follows from the assumption\text{each line follows from the assumption}

    Nothing outside the assumption may be used until the contradiction appears.

  5. Look for a statement of the form P and not P

    P∧¬PP\wedge\lnot P

    A contradiction is any pair of statements that cannot both be true.

  6. Confirm no algebraic slip was made

    re-check each deduction\text{re-check each deduction}

    The contradiction must come from the assumption, not from an error.

  7. Deduce that the assumption cannot hold

    ⇒ assumption is false\Rightarrow\ \text{assumption is false}

    Because it leads to a contradiction, the assumption is impossible.

  8. Invoke the law of the excluded middle

    P∨¬PP\vee\lnot P

    A statement is either true or false, so rejecting the negation proves the claim.

  9. Record the number sets involved

    a,b∈Z, b≠0a,b\in\mathbb{Z},\ b\neq 0

    Being explicit about the sets keeps each deduction valid.

  10. State the defining feature

    assume the negation, then reach a contradiction\text{assume the negation, then reach a contradiction}

    This is what distinguishes proof by contradiction.

Answer
Assuming the statement is false and deriving an impossibility
Question 5
8 markschallenging
Which sequence correctly outlines a proof by contradiction?
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Worked solution

  1. Recall a direct proof

    assume P, deduce Q\text{assume}\ P,\ \text{deduce}\ Q

    A direct proof reasons forward from the hypothesis.

  2. Recall a proof by contradiction

    assume ¬Q, derive P∧¬P\text{assume}\ \lnot Q,\ \text{derive}\ P\wedge\lnot P

    A contradiction proof assumes the negation and reaches an impossibility.

  3. Compare the two approaches

    contradiction targets an impossibility\text{contradiction targets an impossibility}

    The defining feature is the impossibility that is reached.

  4. Work only with consequences of the assumption

    each line follows from the assumption\text{each line follows from the assumption}

    Nothing outside the assumption may be used until the contradiction appears.

  5. Look for a statement of the form P and not P

    P∧¬PP\wedge\lnot P

    A contradiction is any pair of statements that cannot both be true.

  6. Confirm no algebraic slip was made

    re-check each deduction\text{re-check each deduction}

    The contradiction must come from the assumption, not from an error.

  7. Deduce that the assumption cannot hold

    ⇒ assumption is false\Rightarrow\ \text{assumption is false}

    Because it leads to a contradiction, the assumption is impossible.

  8. Invoke the law of the excluded middle

    P∨¬PP\vee\lnot P

    A statement is either true or false, so rejecting the negation proves the claim.

  9. Record the number sets involved

    a,b∈Z, b≠0a,b\in\mathbb{Z},\ b\neq 0

    Being explicit about the sets keeps each deduction valid.

  10. Note the key definition being used

    apply the relevant definition\text{apply the relevant definition}

    The definition of the objects underpins the whole argument.

  11. Summarise the chain of reasoning

    assumption⇒⋯⇒contradiction\text{assumption}\Rightarrow\cdots\Rightarrow\text{contradiction}

    The argument links the assumption directly to an impossibility.

  12. Verify the contradiction is genuine

    contradiction confirmed\text{contradiction confirmed}

    A real contradiction, not an unproved claim, is required to finish.

  13. Restate the established result

    claim now proved\text{claim now proved}

    We restate what has been shown for clarity.

  14. Reflect on why the method succeeds

    eliminating false leaves true\text{eliminating false leaves true}

    Ruling out the only alternative establishes the original statement.

  15. State the defining feature

    assume the negation, then reach a contradiction\text{assume the negation, then reach a contradiction}

    This is what distinguishes proof by contradiction.

Answer
Assume ¬P\lnot P; derive a statement and its negation; conclude PP

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