TMUA: Types of proof and counterexamples
Paper 2: types of proof and counterexamples.
- Section 2 (Paper 2) · Prf1-Prf5 Mathematical proof · spec Prf1
- Paper 2 only
- 8 practice questions
- No calculator
What the specification covers
- Chains of deduction
- Cases such as even/odd or sign of x
- Proof by contradiction (e.g. irrationality, infinitely many primes)
- One counterexample disproves a 'for all' claim
Key ideas
- Direct proof: a chain of implications from what you know to what you want.
- Proof by cases: e.g. \(n\) even and \(n\) odd; together the cases must cover everything.
- Proof by contradiction: assume the statement is false and deduce something impossible.
- One counterexample disproves a 'for all' statement; examples never prove it.
Common mistakes
- Checking a few cases is not a proof.
- A counterexample must satisfy the hypothesis and fail the conclusion.
Exam tip
For 'which is a counterexample' questions, check each option against both halves of the statement.
Worked example
Worked example
The statement 'for every positive integer \(n\), \(n^2-n+11\) is prime' is false. For which of the following values of \(n\) is \(n^2-n+11\) not prime?
- \(n = 9\)
- \(n = 5\)
- \(n = 7\)
- \(n = 11\)
- \(n = 3\)
Answer: D: \(n = 11\)
\(n=11\) gives \(121-11+11=121=11^2\), not prime, so it is a counterexample. The others give \(17, 31, 53, 83\), all prime.
Practice questions
Try each question before opening the solution. No calculator: the TMUA does not allow one. Every answer here was re-checked independently by computer before it was published.
Question 1
Which pair \(a, b\) gives a counterexample to the statement 'if \(a\) and \(b\) are irrational, then \(a+b\) is irrational'?
- \(a=\sqrt{2}\), \(b=\sqrt{3}\)
- \(a=1+\sqrt{2}\), \(b=1-\sqrt{2}\)
- \(a=\pi\), \(b=2\pi\)
- \(a=\sqrt{2}\), \(b=3\sqrt{2}\)
- \(a=\sqrt{8}\), \(b=\sqrt{2}\)
Show the answer and solution
Answer: B: \(a=1+\sqrt{2}\), \(b=1-\sqrt{2}\)
A counterexample needs both irrational and the sum rational. \((1+\sqrt2)+(1-\sqrt2)=2\). The other sums are \(\sqrt2+\sqrt3\), \(4\sqrt2\), \(3\sqrt2\), \(3\pi\), all irrational.
Question 2
The statement 'for every positive integer \(n\), \(n^2-n+17\) is prime' is false. For which of the following values of \(n\) is \(n^2-n+17\) not prime?
- \(n = 13\)
- \(n = 15\)
- \(n = 5\)
- \(n = 17\)
- \(n = 14\)
Show the answer and solution
Answer: D: \(n = 17\)
\(n=17\) gives \(17^2-17+17=289=17^{2}\), so it is not prime: a counterexample. The other values give 199, 227, 37, 173, which are all prime.
Question 3
The statement 'for every positive integer \(n\), \(n^2+n+41\) is prime' is false. For which of the following values of \(n\) is \(n^2+n+41\) not prime?
- \(n = 40\)
- \(n = 7\)
- \(n = 27\)
- \(n = 18\)
- \(n = 24\)
Show the answer and solution
Answer: A: \(n = 40\)
\(n=40\) gives \(40^2+40+41=1681=41^{2}\), so it is not prime: a counterexample. The other values give 97, 383, 641, 797, which are all prime.
Question 4
The statement 'for every positive integer \(n\), \(n^2+n+11\) is prime' is false. For which of the following values of \(n\) is \(n^2+n+11\) not prime?
- \(n = 7\)
- \(n = 10\)
- \(n = 9\)
- \(n = 6\)
- \(n = 2\)
Show the answer and solution
Answer: B: \(n = 10\)
\(n=10\) gives \(10^2+10+11=121=11^{2}\), so it is not prime: a counterexample. The other values give 101, 53, 67, 17, which are all prime.
Question 5
The statement 'for every positive integer \(n\), \(n^2-n+5\) is prime' is false. For which of the following values of \(n\) is \(n^2-n+5\) not prime?
- \(n = 4\)
- \(n = 5\)
- \(n = 3\)
- \(n = 1\)
- \(n = 2\)
Show the answer and solution
Answer: B: \(n = 5\)
\(n=5\) gives \(5^2-5+5=25=5^{2}\), so it is not prime: a counterexample. The other values give 7, 17, 5, 11, which are all prime.
Question 6
The statement 'for every positive integer \(n\), \(n^2-n+41\) is prime' is false. For which of the following values of \(n\) is \(n^2-n+41\) not prime?
- \(n = 26\)
- \(n = 17\)
- \(n = 27\)
- \(n = 13\)
- \(n = 41\)
Show the answer and solution
Answer: E: \(n = 41\)
\(n=41\) gives \(41^2-41+41=1681=41^{2}\), so it is not prime: a counterexample. The other values give 313, 743, 197, 691, which are all prime.
Question 7
The statement 'for every positive integer \(n\), \(n^2+n+17\) is prime' is false. For which of the following values of \(n\) is \(n^2+n+17\) not prime?
- \(n = 14\)
- \(n = 15\)
- \(n = 5\)
- \(n = 2\)
- \(n = 16\)
Show the answer and solution
Answer: E: \(n = 16\)
\(n=16\) gives \(16^2+16+17=289=17^{2}\), so it is not prime: a counterexample. The other values give 257, 47, 227, 23, which are all prime.
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