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Date November 2013 Marks available 9 Reference code 13N.3srg.hl.TZ0.4
Level HL only Paper Paper 3 Sets, relations and groups Time zone TZ0
Command term Determine and Show that Question number 4 Adapted from N/A

Question

Let (H, ) be a subgroup of the group (G, ).

Consider the relation R defined in G by xRy if and only if y1xH.

(a)     Show that R is an equivalence relation on G.

(b)     Determine the equivalence class containing the identity element.

Markscheme

(a)     R is reflexive as x1x=eHxRx for any xG     A1

if xRy then y1x=hH

but hHh1H, ie(y1x)1x1*yH     M1

therefore yRx

R is symmetric     A1

if xRy then y1x=hH and if yRz then z1y=kH     M1

khH, ie, (z1y)(y1x)z1xH     A1

therefore xRz

R is transitive     A1

so R is an equivalence relation on G     AG

[6 marks]

(b)     xRee1xH     M1

xH     A1

[e]=H     A1     N0

[3 marks]

Examiners report

Part (a) was fairly well answered by many candidates. They knew how to apply the equivalence relations axioms in this particular example. Part (b) however proved to be very challenging and hardly any correct answers were seen.

Syllabus sections

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