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Date May 2012 Marks available 2 Reference code 12M.3srg.hl.TZ0.3
Level HL only Paper Paper 3 Sets, relations and groups Time zone TZ0
Command term Find Question number 3 Adapted from N/A

Question

The relation R is defined on the set \(\mathbb{N}\) such that for \(a{\text{ }},{\text{ }}b \in \mathbb{N}{\text{ }},{\text{ }}aRb\) if and only if \({a^3} \equiv {b^3}(\bmod 7)\).

Show that R is an equivalence relation.

[6]
a.

Find the equivalence class containing 0.

[2]
b.

Denote the equivalence class containing n by Cn .

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

[3]
c.

Denote the equivalence class containing n by Cn .

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

[3]
d.

Markscheme

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]

a.

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

[2 marks]

b.

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

Note: Deduct 1 mark for each error or omission.

 

[3 marks]

c.

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]

d.

Examiners report

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.

a.

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.

b.

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.

c.

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.

d.

Syllabus sections

Topic 8 - Option: Sets, relations and groups » 8.2 » Relations: equivalence relations; equivalence classes.
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