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Date May 2015 Marks available 2 Reference code 15M.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 Z by xRy if and only if x2yymod6.

Show that the product of three consecutive integers is divisible by 6.

[2]
a.

Hence prove that R is reflexive.

[3]
b.

Find the set of all y for which 5Ry.

[3]
c.

Find the set of all y for which 3Ry.

[2]
d.

Using your answers for (c) and (d) show that R is not symmetric.

[2]
e.

Markscheme

in a product of three consecutive integers either one or two are even     R1

and one is a multiple of 3     R1

so the product is divisible by 6     AG

[2 marks]

a.

to test reflexivity, put y=x     M1

then x2xx=(x1)x(x+1)0mod6     M1A1

so xRx     AG

[3 marks]

b.

if 5Ry then 25yymod6     (M1)

24y0mod6     (M1)

the set of solutions is Z     A1

 

Note:     Only one of the method marks may be implied.

[3 marks]

c.

if 3Ry then 9yymod6

8y0mod64y0mod3     (M1)

the set of solutions is 3Z (ie multiples of 3)     A1

[2 marks]

d.

from part (c) 5R3     A1

from part (d) 3R5 is false     A1

R is not symmetric     AG

 

Note:     Accept other counterexamples.

[2 marks]

Total [12 marks]

e.

Examiners report

A surprising number of candidates thought that an example was sufficient evidence to answer this part.

a.

Again, a lack of confidence with modular arithmetic undermined many candidates' attempts at this part.

b.

(c) and (d) Most candidates started these parts, but some found solutions as fractions rather than integers or omitted zero and/or negative integers.

c.

(c) and (d) Most candidates started these parts, but some found solutions as fractions rather than integers or omitted zero and/or negative integers.

d.

Some candidates regarded R as an operation, rather than a relation, so returned answers of the form aRbbRa.

e.

Syllabus sections

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