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.