The relation aRb is defined on {1, 2, 3, 4, 5, 6, 7, 8, 9} if and only if ab is the square of a positive integer. 

(i)     Show that R is an equivalence relation. 

(ii)     Find the equivalence classes of R that contain more than one element.

Given the group \((G,{\text{ }} * )\), a subgroup \((H,{\text{ }} * )\) and \(a,{\text{ }}b \in G\), we define \(a \sim b\) if and only if \(a{b^{ - 1}} \in H\). Show that \( \sim \) is an equivalence relation.

(i)     \(aRa \Rightarrow a \cdot a = {a^2}\) so R is reflexive     A1

\(aRb = {m^2} \Rightarrow bRa\) so R is symmetric     A1

\(aRb = ab = {m^2}{\text{ and }}bRc = bc = {n^2}\)     M1A1

so \(a = \frac{{{m^2}}}{b}{\text{ and }}c = \frac{{{n^2}}}{b}\)

\(ac = \frac{{{m^2}{n^2}}}{{{b^2}}} = {\left( {\frac{{mn}}{b}} \right)^2},\)     A1

ac is an integer hence \({\left( {\frac{{mn}}{b}} \right)^2}\) is an integer     R1

so aRc, hence R is transitive     R1

R is therefore an equivalence relation     AG

(ii)     1R4 and 4R9 or 2R8     M1

so {1, 4, 9} is an equivalence class     A1

and {2, 8} is an equivalence class     A1

[10 marks]

\(a \sim a{\text{ since }}a{a^{ - 1}} = e \in H\), the identity must be in H since it is a subgroup.     M1

Hence reflexivity.     R1

\(a \sim b \Leftrightarrow a{b^{ - 1}} \in H\) but H is a subgroup so it must contain \({(a{b^{ - 1}})^{ - 1}} = b{a^{ - 1}}\)     M1R1

i.e. \(b{a^{ - 1}} \in H{\text{ so }} \sim \) is symmetric     A1

\(a \sim b{\text{ and }}b \sim c \Rightarrow a{b^{ - 1}} \in H{\text{ and }}b{c^{ - 1}} \in H\)     M1

But H is closed, so

\((a{b^{ - 1}})(b{c^{ - 1}}) \in H{\text{ or }}a({b^{ - 1}}b){c^{ - 1}} \in H\)     R1

\(a{c^{ - 1}} \in H \Rightarrow a \sim c\)     A1

Hence \( \sim \) is transitive and is thus an equivalence relation     R1AG

[9 marks]

Not a difficult question although using the relation definition to fully show transitivity was not well done. It was good to see some students use an operation binary matrix to show transitivity. This was a nice way given that the set was finite. The proof in (b) proved difficult.

Not a difficult question although using the relation definition to fully show transitivity was not well done. It was good to see some students use an operation binary matrix to show transitivity. This was a nice way given that the set was finite. The proof in (b) proved difficult.