Show that $R$ is an equivalence relation.

State the equivalence classes of $R$ .

$4a + b = 5n$ for $a,b,n \in \mathbb{Z}$

reflexive:

$4a + a = 5a$ so $aRa$ , and $R$ is reflexive     A1

symmetric:

$4a + b = 5n$

$4b + a = 5b - b + 5a - 4a$     M1

$= 5b + 5a - (4a + b)$     A1

$= 5m$ so $bRa$ , and $R$ is symmetric     A1

transitive:

$4a + b = 5n$     M1

$4b + c = 5k$     M1

$4a + 5b + c = 5n + 5k$     A1

$4a + c = 5(n + k - b)$ so $aRc$ , and $R$ is transitive     A1

therefore $R$ is an equivalence relation     AG

[8 marks]

equivalence classes are

$\left\{ { \ldots , - 10, - 5,0,5,10,\left.  \ldots  \right\}} \right.$     (M1)

$\left\{ { \ldots , - 9, - 4,1,6,11,\left.  \ldots  \right\}} \right.$

$\left\{ { \ldots , - 8, - 3,2,7,12,\left.  \ldots  \right\}} \right.$

$\left\{ { \ldots , - 7, - 2,3,8,13,\left.  \ldots  \right\}} \right.$

$\left\{ { \ldots , - 6, - 1,4,9,14,\left.  \ldots  \right\}} \right.$

or $\left\{ {\left\langle 0 \right\rangle ,\left\langle 1 \right\rangle ,\left\langle 2 \right\rangle ,\left\langle 3 \right\rangle ,\left. {\left\langle 4 \right\rangle } \right\}} \right.$     A2

Note: Award A2 for all classes, A1 for at least 2 correct classes.

[3 marks]

Part (a) was generally well done but not always in the most direct manner.

Too many missed the equivalence classes in part (b).