dmat hand-in

8/12/2019 DMAT Hand-in

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Christoffer Werge (10832) Hand-in 3 13-09-2013

Lasse Srensen (09421)

Hand-in 3

Exercise 1

We are told to disproe the state!ent"

If x {0,1,2,3,4}, then 2 n+ 3n+ n*(n-1)* (n-2) is prime

#he $est %a& to disproe this senten'e is $& giing a 'onterea!ple in the for! of"

There exists an x {0,1,2,3,4}, such that 2 n+ 3n+ n*(n-1)* (n-2) is not prime

We find ot* that %hen is 4 the e+ation gies s"

24 + 34 + 4*(4-1) * (4-2) 1! + "1 + 24 121

11 is a fa'tor of 121* and therefore 121 is not a pri!e, So %e hae fond a 'onterea!ple to orfirst state!ent* and ths disproed it,

Exercise 2

. is odd if $oth and . is odd* is een if * . or .oth is een" 3 og / 1/

( .)2* is odd if and . is of different parit&* is een if and . is of sa!e parit&, 3 og / 4

2.2 is odd if $oth and . is odd* is een if * . or $oth are een, 3 og / 22/ odd

. .* is odd if * . or .oth are odd* is een if .oth are een, 3 og / 23 odd

We are told that and . is in the do!ain of 5, We are then as6ed to disproe the state!ent"

If #*$ an% (# + $)2 are of opposite parit&,

then #2 $2 an% # + #*$ + $ are of opposite parit&

#o disproe this* %e need to find atleast one and one . %here this doesn7t hold, n anal&sing the

indiidal stat!ents %e 'an figre ot that"

#*$ is o%% iff' oth # an% $ are o%%'

(#+$)2 is o%% iff' # an% $ is of %ifferent parit&'

#2*$2 is o%% iff' $oth # an% $ are o%%'

# + #*$ + $ is o%% iff' #, $ or $oth are o%%'

ro! this %e 'an see that"

#*$ an% #2*$2 are aa&s of the same parit&'

f %e 'an find an and a . that !a6es (.)2 a different parit& fro! . and . .* then

%e hae disproed the state!ents, We 'hoose and and . that $oth are odd, #his %old !a6e .

odd and (.)2 een* 2.2 %old $e odd and . . is odd, So"

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If # an% $ is o%%, then #*$ an% (#*$) is of %ifferent parit&,

an% #2*$2 an% # + #*$ + $ is of the same parit&

Exercise 3We are told that and & is in do!ain of all positie real n!$ers,

rther!ore %e are told to proe $& 'ontradi'tion that if : & then s+rt() : s+rt(&)

#o proe this $& 'ontradi'tion %e ass!e to the 'ontrar&* that"there exists x an% & such that if x & then srt(x) srt(&)

We ass!e that a* and & a 1* a $eing a real n!$er, Whateer n!$er &o pt into a*

%old al%a&s $e s!aller then &* So"

if a a + 1, then srt(a) srt(a+1)

When sing real n!$ers* %e 'an safel& ass!e that"

if s t, then s2 t2

So %e 'an ass!e that"

If a a+1, then srt(a)2

srt(a+1)2

.

if a a+1, then a a+1

#his is a 'lear 'ontradi'tion* and so it is proed $& 'ontradi'tion that"

x & then srt(x) srt(&)

/''

Exercise 4:

We need to proe that there is no largest negatie rational n!$er,#o do this %e ass!e to the 'ontrar& that there is a largest negatie n!$er* %hi'h %e %ill 'all s, t

!st follo% that"

/- s 0

f s is rational* negatie n!$er* then there is a (er&* er& s!all) n!$er $et%een s and 0, Lets 'all

this differen'e t, #he half of this distan'e* 'an $e 'alled t;2, #his distan'e added to s* %old still $e

negatie* $t greater than s* so"

/- s s + t2 0

#herefore s is not the greatest negatie n!$er* %hi'h %ill 'reate a 'ontradi'tion, So it has $een

proen $& 'ontradi'tion that there is not largest negatie n!$er

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Exercise 5:

We need to proe that there eists no positie integer s'h that 2 : 2 : 3,

Sin'e is a positie integer* %e 'an safel& ass!e that

a c . ax x cx

#his !st !ean %e 'an sa& that"

2x x x2 x 3x x .

2 x 3

Sin'e there 'an7t eist a positie integer $et%een 2 and 3* %e hae proed that there eists no

positie integer s'h that 2 : 2: 3,

Exercise 6:

We need to proe that if n is an odd integer* then ?n @ / is een* $& doing 3 different proofsA >ire'tproof* proof $& 'ontrapositie and proof $& 'ontradi'tion

art # irect roof'

f n is odd* %e 'an s$titte this $& sa&ing that n 2 1, ro! this %e 'an ded'e that"

5(2x +1) 6 7

14 x + 5 6 7

14x + 2

2(5x + 1)

Sin'e ? 1 is an integer* ?n @ / is een,

art $ roof & contrapositi8e

#o do a proof $& 'ontrapositie %e sa& that if ?n @ / is odd* then n is een, We start ot $&

ass!ing that n is een* and 'an $e s$stitted %ith 2, ro! this %e 'an ded'e that"

5(2x) 6 7

14x 6 7

14 x 6 4 6 1

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2(5x 6 2) 6 1

Sin'e ? @ 2 is an integer* ?n @ / is odd, Sin'e the state!ent Bf ?n @ / is odd* then n is eenB is

logi'all& e+ialent to Bf n is odd* then ?n @ / is eenB* %e hae proed the state!ent sing proof

$& 'ontrapositie,

art 9 roof & contra%iction'

We %ill no% ass!e to the 'ontrar& that there eists and een integer n* s'h that ?n-/ is een,

Sin'e n is een* %e 'an s$stitte n $& 2, ro! this %e 'an ded'e that"

5(2x) 6 7

14x 6 7

14 x 6 4 6 12(5x 6 2) 6 1

Sin'e ? @ 2 is an integer* ?n-/ is odd, #his 'ontradi'ts or preios state!ent* and ths the reerse*

that if n is een* then ?n -/ is odd* !st $e tre, #herefore it has $een proed $& 'ontradi'tion

Exercise 7:

We hae to sho% that there eists t%o different irrational n!$ers and . s'h that . is rational,

We %ill no% 'onsider the irrational n!$ers s+rt(2) and . 2

s+rt(2)

,

$

%old then $e 4, So t%odistin't irrational n!$ers* 'an prod'e a rational n!$er,

Exercise 8:

We need to disproe the state!ent" #here is an integer n s'h that n4 n3 n2 n is odd, We %ill do

this sing a proof $& 'ases"

9ase 1 n is e8en

f n is een it 'an $e s$stitted $& 2, ro! this %e 'an ded'e that"

(2x)4 + (2x)3 + (2x)2 + (2x)

2(x4+ x3 + x2 + x)

Sin'e x4+ x3 + x2 + x is an integer* it !st follo% that n4 n3 n2 n is een for the 'ase that n is

een, #hs no een n* 'an !a6e n4 n3 n2 n odd,

9ase 1 n is o%%

f n is odd it 'an $e s$stitted $& 2 1 , ro! this %e 'an ded'e that"

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(2x + 1)4 + (2x +1)3 + (2x +1)2 + (2x +1)

1!x4 + 40x3+ 40x2 + 1"x + 4

2("x4

+ 20x3

+ 20x2

+ :x + 2)Sin'e "x4+ 20x3 + 20x2+ :x + 2is an integer it !st follo% that n4 n3 n2 n is een for the 'ase

that n is odd, #hs no odd n* 'an !a6e n4 n3 n2 n odd,

o %e hae tried %ith $oth n $eing een and n $eing odd* and %e 'old prod'e no odd n!$er, So

it !st follo% that %e hae disproed the senten'e" #here is an integer n s'h that n4 n3 n2 n is

odd

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