Show that R is an equivalence relation.

Find the equivalence class containing 0.

Denote the equivalence class containing n by C_n .

List the first six elements of \({C_1}\).

Denote the equivalence class containing n by C_n .

Prove that \({C_n} = {C_{n + 7}}\) for all \(n \in \mathbb{N}\).

reflexive: \({a^3} - {a^3} = 0{\text{ }},{\text{ }} \Rightarrow R\) is reflexive     R1

symmetric: if \({a^3} \equiv {b^3}(\bmod 7)\) , then \({b^3} \equiv {a^3}(\bmod 7)\)     M1

\( \Rightarrow R\) is symmetric     R1

transitive: \({a^3} = {b^3} + 7n\) and \({b^3} = {c^3} + 7m\)     M1

then \({a^3} = {c^3} + 7(n + m)\)

\( \Rightarrow {a^3} \equiv {c^3}(\bmod 7)\)     R1

\( \Rightarrow R\) is transitive     A1

and is an equivalence relation     AG 

Note: Allow arguments that use \({a^3} - {b^3} \equiv 0(\bmod 7)\) etc.

[6 marks]

\(\{ 0,{\text{ }}7,{\text{ }}14,{\text{ }}21,{\text{ }}...\} \)     A2

[2 marks]

\(\{ 1,{\text{ }}2,{\text{ }}4,{\text{ }}8,{\text{ }}9,{\text{ }}11\} \)     A3 

Note: Deduct 1 mark for each error or omission.

[3 marks]

consider \({(n + 7)^3} = {n^3} + 21{n^2} + 147n + 343 = {n^3} + 7N\)     M1A1

\( \Rightarrow {n^3} \equiv {(n + 7)^3}(\bmod 7) \Rightarrow n\) and \(n + 7\) are in the same equivalence class     R1

[3 marks]

Candidates were mostly aware of the conditions required to show an equivalence relation although many seemed unsure as to the degree of detail required to show that the different conditions are met for the example in this question. In part (b) many candidates found the correct set although a number were unable to write down the set correctly, including or excluding elements that were not part of the equivalence class. Part (c) saw candidate being less successful than (b) and relatively few candidates were able to prove the equivalence class in part (d) although there were a number of very good solutions.

Candidates were mostly aware of the conditions required to show an equivalence relation although many seemed unsure as to the degree of detail required to show that the different conditions are met for the example in this question. In part (b) many candidates found the correct set although a number were unable to write down the set correctly, including or excluding elements that were not part of the equivalence class. Part (c) saw candidate being less successful than (b) and relatively few candidates were able to prove the equivalence class in part (d) although there were a number of very good solutions.

Candidates were mostly aware of the conditions required to show an equivalence relation although many seemed unsure as to the degree of detail required to show that the different conditions are met for the example in this question. In part (b) many candidates found the correct set although a number were unable to write down the set correctly, including or excluding elements that were not part of the equivalence class. Part (c) saw candidate being less successful than (b) and relatively few candidates were able to prove the equivalence class in part (d) although there were a number of very good solutions.

Candidates were mostly aware of the conditions required to show an equivalence relation although many seemed unsure as to the degree of detail required to show that the different conditions are met for the example in this question. In part (b) many candidates found the correct set although a number were unable to write down the set correctly, including or excluding elements that were not part of the equivalence class. Part (c) saw candidate being less successful than (b) and relatively few candidates were able to prove the equivalence class in part (d) although there were a number of very good solutions.