WJEC Mathematics for AS Level - Pure

1.4 Disproof by counter-example 11 Examples 1 Prove that if n is an integer between and including 1 and 5 (i.e. 1 ≤ n ≤ 5), then the expression n 2 − n + 11 is prime. Answer 1 As there are a limited number of values we can use with the expression, we can enter them in turn into the expression and check that the result is a prime number. When n = 1, n 2 − n + 11 = 1 2 − 1 + 11 = 11 n = 2, n 2 − n + 11 = 2 2 − 2 + 11 = 13 n = 3, n 2 − n + 11 = 3 2 − 3 + 11 = 17 n = 4, n 2 − n + 11 = 4 2 − 4 + 11 = 23 n = 5, n 2 − n + 11 = 5 2 − 5 + 11 = 31 All the results are prime numbers, so the statement is true. 2 If p and q are even integers less than 6, prove that the sum and difference of p and q are divisible by two. Answer 2 Possible values for p are 2 and 4 and possible values for q are also 2 and 4. We can construct a table to list all the possible values: p q p + q p − q Divisible by 2? 2 2 4 0 Yes 2 4 6 −2 Yes 4 2 6 2 Yes 4 4 8 0 Yes As all the possible values have been exhausted and all the answers are divisible by 2, the statement is correct. 1.4 Disproof by counter-example Disproof by counter-example allows us to prove that a property is not true by providing an example where it does not hold. For example, to disprove that ‘all triangles are obtuse’, we give the following disproof by counter-example: the equilateral triangle having all angles equal to sixty degrees. There are an inϐinite number of counter-examples we could have used here, but it only takes one of them to disprove this particular statement. N.B. A conjecture involving the word ‘all’ cannot be proved by showing that it is true in one case; but it can be disproved by showing that it is untrue in one case. Note that by entering all the possible values we have exhausted all the values. Hence if all the results are prime, then the statement is true.