Consider the following statements

$z\,:\,x$ is an integer
$q\,:\,x$ is a rational number
$r\,:\,x$ is a real number.

i)    Write down, in words, $\neg q$.

ii)   Write down a value for $x$ such that the statement $\neg q$ is true.

Write the following argument in symbolic form:
“If $x$ is a real number and $x$ is not a rational number, then $x$ is not an integer”.

Phoebe states that the argument in part (b) can be shown to be valid, without the need of a truth table.

Justify Phoebe’s statement.

i)    $x$ is not a rational number        (A1)

Note: Accept “$x$ is an irrational number”.

ii)   any non-rational number (for example: $\pi ,\,\sqrt 2$, …)       (A1) (C2)

$(r \wedge \neg q) \Rightarrow \neg z$       (A1)(A1)(A1) (C3)

Note: Award (A1) for “$\Rightarrow$” seen, (A1) for “$\neg z$” as the consequent and (A1) for “$(r \wedge \neg q)$” or “$(\neg q \wedge r)$” as the antecedent (the parentheses are required).

all integers are rational numbers (and therefore $x$ cannot be an integer if it is not a rational number)       (R1)

Note: Accept equivalent expressions.

OR

if $x$ is an integer, then $x$ is a rational number, therefore if $x$ is not a rational number, then $x$ is not an integer (contrapositive)   (R1) (C1)

Note: Accept “If $x$ is not in $\mathbb{Q}$, then $x$ is not in $\mathbb{Z}$” with a Venn diagram showing $\mathbb{R}$, $\mathbb{Q}$ and $\mathbb{Z}$ correctly.

Question 5 Logic
In part (a), the majority of candidates were able to state the negation, but surprisingly many were unable to give an example of a non-rational number.

In part (b), a common error was the lack of parentheses in the antecedent. A further error was the use of the “intersection” symbol rather than that for conjunction; care must be taken in this regard.

Part (c) proved problematic for all but the best candidates.