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Date November 2010 Marks available 8 Reference code 10N.3srg.hl.TZ0.1
Level HL only Paper Paper 3 Sets, relations and groups Time zone TZ0
Command term Determine and Write down Question number 1 Adapted from N/A

Question

Let R be a relation on the set Z such that aRbab, for a, b \in \mathbb{Z}.

(a)     Determine whether R is

(i)     reflexive;

(ii)     symmetric;

(iii)     transitive.

(b)     Write down with a reason whether or not R is an equivalence relation.

Markscheme

(a)     (i)     {a^2} \geqslant 0 for all a \in \mathbb{Z}, hence R is reflexive     R1

 

(ii)     aRb \Rightarrow ab \geqslant 0     M1

\Rightarrow ba \geqslant 0     R1

\Rightarrow bRa, hence R is symmetric     A1

 

(iii)     aRb{\text{ and }}bRc \Rightarrow ab \geqslant 0{\text{ and }}bc \geqslant 0,{\text{ is }}aRc?     M1

no, for example, - 3R0 and 0R5, but - 3R5 is not true     A1

aRc is not generally true, hence R is not transitive     A1

[7 marks]

 

(b)     R does not satisfy all three properties, hence R is not an equivalence relation     R1

[1 mark]

Total [8 marks]

Examiners report

Although the properties of an equivalence relation were well known, few candidates provided a counter-example to show that the relation is not transitive. Some candidates interchanged the definitions of the reflexive and symmetric properties.

Syllabus sections

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