Show that R is an equivalence relation.

Determine the equivalence class of R containing the element $(1,{\text{ }}2)$ and illustrate this graphically.

R is an equivalence relation if

R is reflexive, symmetric and transitive     A1

${x_1}{y_1} = {x_1}{y_1} \Rightarrow ({x_1},{\text{ }}{y_1})R({x_1},{\text{ }}{y_1})$     A1

so R is reflexive

$({x_1},{\text{ }}{y_1})R({x_2},{\text{ }}{y_2}) \Rightarrow {x_1}{y_1} = {x_2}{y_2} \Rightarrow {x_2}{y_2} = {x_1}{y_1} \Rightarrow ({x_2},{\text{ }}{y_2})R({x_1},{\text{ }}{y_1})$     A1

so R is symmetric

$({x_1},{\text{ }}{y_1})R({x_2},{\text{ }}{y_2})$ and $({x_2},{\text{ }}{y_2})R({x_3},{\text{ }}{y_3}) \Rightarrow {x_1}{y_1} = {x_2}{y_2}$ and ${x_2}{y_2} = {x_3}{y_3}$     M1

$\Rightarrow {x_1}{y_1} = {x_3}{y_3} \Rightarrow ({x_1},{\text{ }}{y_1})R({x_3},{\text{ }}{y_3})$    A1

so R is transitive

R is an equivalence relation     AG

[5 marks]

$(x,{\text{ }}y)R(1,{\text{ }}2)$     (M1)

the equivalence class is $\{ (x,{\text{ }}y)|xy = 2\}$     A1

correct graph     A1

$(1,{\text{ }}2)$ indicated on the graph     A1

Note:     Award last A1 only if plotted on a curve representing the class.

[4 marks]