TMUA Practice Question — Worked Solution

P2·Sets & Counting·7.0Mock 0 · P2 · Q15

sum-free setmaximal versus largestpairing argumentproof by contradictioncounterexample

A set of positive integers is sum-free if there are no elements

x

y

z

of the set, not necessarily different, with

x + y = z

. For example,

{2, 3, 7}

is sum-free, but

{2, 4, 5}

is not, because

2 + 2 = 4

.A sum-free subset of

{1, 2, \dots, 10}

is full if adding any other element of

{1, 2, \dots, 10}

to it would stop it being sum-free.Let

M

be the largest possible number of elements in a sum-free subset of

{1, 2, \dots, 10}

Every full sum-free subset of

{1, 2, \dots, 10}

has

M

elements.

Every sum-free subset of

{1, 2, \dots, 10}

with

M

elements contains

1

10

III

Every sum-free subset of

{1, 2, \dots, 10}

with

M

elements contains

9

Which of these statements is/are true?

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