(i)     Write down the six smallest non-negative elements of $S$.

(ii)     Show that $\{ S,{\text{ }} + \}$ is a group.

(iii)     Give a reason why $\{ S,{\text{ }} \times \}$ is not a group. Justify your answer.

The relation $R$ is defined on $S$ by ${s_1}R{s_2}$ if $3{s_1} + 5{s_2} \in \mathbb{Z}$.

(i)     Show that $R$ is an equivalence relation.

(ii)     Determine the equivalence classes.

(i)     ${\text{0, }}\frac{{\text{1}}}{{\text{2}}}{\text{, 1, }}\frac{{\text{3}}}{{\text{2}}}{\text{, 2, }}\frac{{\text{5}}}{{\text{2}}}$     A2

Notes:     A2 for all correct, A1 for three to five correct.

(ii)     EITHER

closure: if ${s_1},{\text{ }}{s_2} \in S$, then ${s_1} = \frac{m}{2}$ and ${s_2} = \frac{n}{2}$ for some $m,{\text{ }}n \in {\text{¢}}$.     M1

Note:     Accept two distinct examples (eg, $\frac{1}{2} + \frac{1}{2} = 1;{\text{ }}\frac{1}{2} + 1 = \frac{3}{2}$) for the M1.

${s_1} + {s_2} = \frac{{m + n}}{2} \in S$     A1

OR

the sum of two half-integers     A1

is a half-integer     R1

THEN

identity: 0 is the (additive) identity     A1

inverse: $s + ( - s) = 0$, where $- s \in S$     A1

it is associative (since $S \subset \S$)     A1

the group axioms are satisfied     AG

(iii)     EITHER

the set is not closed under multiplication,     A1

for example, $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$, but $\frac{1}{4} \notin S$     R1

OR

not every element has an inverse,     A1

for example, 3 does not have an inverse     R1

[9 marks]

(i)     reflexive: consider $3s + 5s$     M1

$= 8s \in {\text{¢}} \Rightarrow$ reflexive     A1

symmetric: if ${s_1}R{s_2}$, consider $3{s_2} + 5{s_1}$     M1

for example, $= 3{s_1} + 5{s_2} + (2{s_1} - 2{s_2}) \in {\text{¢}} \Rightarrow$symmetric     A1

transitive: if ${s_1}R{s_2}$ and ${s_2}R{s_3}$, consider     (M1)

$3{s_1} + 5{s_3} = (3{s_1} + 5{s_2}) + (3{s_2} + 5{s_3}) - 8{s_2}$     M1

$\in {\text{¢}} \Rightarrow$transitive     A1

so R is an equivalence relation     AG

(ii)     ${C_1} = {\text{¢}}$     A1

${C_2} = \left\{ { \pm \frac{1}{2},{\text{ }} \pm \frac{3}{2},{\text{ }} \pm \frac{5}{2},{\text{ }} \ldots } \right\}$     A1A1

Note: A1 for half odd integers and A1 for ±.

[10 marks]