The relation R is defined on ${\mathbb{Z}^ + }$ by aRb if and only if ab is even. Show that only one of the conditions for R to be an equivalence relation is satisfied.

The relation S is defined on ${\mathbb{Z}^ + }$ by aSb if and only if ${a^2} \equiv {b^2}(\bmod 6)$ .

(i)     Show that S is an equivalence relation.

(ii)     For each equivalence class, give the four smallest members.

reflexive: if a is odd, $a \times a$ is odd so R is not reflexive     R1

symmetric: if ab is even then ba is even so R is symmetric     R1

transitive: let aRb and bRc; it is necessary to determine whether or not aRc     (M1)

for example 5R2 and 2R3     A1

since $5 \times 3$ is not even, 5 is not related to 3 and R is not transitive     R1

[5 marks]

(i)     reflexive: ${a^2} \equiv {a^2}(\bmod 6)$ so S is reflexive     R1

symmetric: ${a^2} \equiv {b^2}(\bmod 6) \Rightarrow 6|({a^2} - {b^2}) \Rightarrow 6|({b^2} - {a^2}) \Rightarrow {b^2} \equiv {a^2}(\bmod 6)$     R1

so S is symmetric

transitive: let aSb and bSc so that ${a^2} = {b^2} + 6M$ and ${b^2} = {c^2} + 6N$     M1

it follows that ${a^2} = {c^2} + 6(M + N)$ so aSc and S is transitive     R1

S is an equivalence relation because it satisfies the three conditions     AG

(ii)     by considering the squares of integers (mod 6), the equivalence     (M1)

classes are

{1, 5, 7, 11, $\ldots$}     A1

{2, 4, 8, 10, $\ldots$}     A1

{3, 9, 15, 21, $\ldots$}     A1

{6, 12, 18, 24, $\ldots$}     A1

[9 marks]