Let R be a relation on the set \(\mathbb{Z}\) such that \(aRb \Leftrightarrow ab \geqslant 0\), for a, b \( \in \mathbb{Z}\).

(a)     Determine whether R is

(i)     reflexive;

(ii)     symmetric;

(iii)     transitive.

(b)     Write down with a reason whether or not R is an equivalence relation.

(a)     (i)     \({a^2} \geqslant 0\) for all \(a \in \mathbb{Z}\), hence R is reflexive     R1

(ii)     \(aRb \Rightarrow ab \geqslant 0\)     M1

\( \Rightarrow ba \geqslant 0\)     R1

\( \Rightarrow bRa\), hence R is symmetric     A1

(iii)     \(aRb{\text{ and }}bRc \Rightarrow ab \geqslant 0{\text{ and }}bc \geqslant 0,{\text{ is }}aRc?\)     M1

no, for example, \( - 3R0\) and \(0R5\), but \( - 3R5\) is not true     A1

aRc is not generally true, hence R is not transitive     A1

[7 marks]

(b)     R does not satisfy all three properties, hence R is not an equivalence relation     R1

[1 mark]

Total [8 marks]

Although the properties of an equivalence relation were well known, few candidates provided a counter-example to show that the relation is not transitive. Some candidates interchanged the definitions of the reflexive and symmetric properties.