WJEC Mathematics for AS Level - Pure

1 Proof 12 Examples 1 Here is a statement: ‘for all real numbers x , if x 2 is rational, then x is rational’. By giving a counter-example, disprove this statement. Answer 1 Suppose x = √ 2 (which is an irrational number) x 2 = ( √ 2 ) 2 = 2 (which is a rational number) We have found a counter-example where x 2 is rational but the value of x is irrational. This means that the statement is not true for all real values of x . 2 Disprove the following mathematical statement using a suitable counter- example. The graphs of all linear functions in a single variable are perpendicular to each other. Answer 2 Let one function be f  ( x ) = 2 x − 1 and the other function be g ( x ) = −3 x + 2. These functions are both in the form y = mx + c . The gradient of f  ( x ) is 2 whilst the gradient of g ( x ) is −3. For two straight lines to be perpendicular to each other, the product of the gradients must equal −1. Product of gradients in this case = 2 × (−3) = −6. This proves that not all linear functions are perpendicular to each other. Hence the given mathematical statement is incorrect. 3 Show, by counter-example, that the following statement is false. ‘If the integers a , b , c , d are such that a is a factor of c and b is a factor of d , then ( a + b ) is a factor of ( c + d ). Answer 3 Suppose a = 3 and c = 18 (so a is a factor of c ) and suppose b = 2 and d = 10 (so that b is a factor of d ). ( a + b ) = 3 + 2 = 5 and ( c + d ) = 18 + 10 = 28 5 is not a factor of 28 so ( a + b ) is not a factor of ( c + d ) for this set of numbers. Hence the statement is false. 4 Disprove the following using an appropriate counter-example. If in the quadratic equation ax 2 + bx + c = 0, the numbers a , b and c are real, then if b is negative, the roots of the equation will be negative. Remember an irrational number cannot be expressed exactly, so you cannot locate its exact position on a number line. Irrational numbers have numbers after the decimal point which do not repeat and cannot be expressed as a fraction.