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MID SWEDEN UNIVERSITY DMA

Examination 2014

MA095G & MA098G Discrete Mathematics (English)

Time: 5 hours

Date: 19 March 2014

Pia Heidtmann

The compulsory part of this examination consists of 9 questions. The maximum number of points available is 24. The points for each part of a question are indicated at the end of the part in [ ]-brackets.

The final grade on the course is determined by how well the candidates demonstrate that they have met the learning outcomes on the course. Provided all learning outcomes have been met, the following guide values will be used to set the course grade:

E: 9p D: 10p C: 14p B: 18p A: 22p

The final question on the paper is the Aspect Question, it is optional and carries no value in terms of marks, but a good solution of this Aspect Question may raise a candidates grade by one grade.

The candidates are advised that they must always show their working, otherwise they will not be awarded full marks for their answers. The candidates are further advised to start each of the nine questions on a new page and to clearly label all their answers.

This is a closed book examination. No books, notes or mobile telephones are allowed in the examination room. Note that Mathematical Formula Collection Edition 3 is allowed on this tenta and will be available in the examination room.

Electronic calculators may be used provided they cannot handle formulas.

The make and model used must be specified on the cover of your script.

GOOD LUCK!!

DMA, Mid Sweden Universityc 1 MA095G & MA098G

(2)

Question 1

(a) Express the hexadecimal number 20C in base 10. [0.5p]

(b) Compute the product [2]−1 [5] in Z7. [0.5p]

(c) Write the sum

s = 1· 2 · 3 + 4 · 5 · 6 + 7 · 8 · 9 + . . . + 199 · 200 · 201

by using Σ notation. [1p]

(d) Showing all your working, compute the sum

2014X

n=193

(4n − 3).

[1p]

Question 2

(a) Let p and q be logical propositions. Give a truth table for the following two composite propositions and decide whether they are equivalent or not.

p1: p ⇒ q;

p2: (¬p) ⇒ (¬q). [1p]

(b) Consider the following statement concerning an integer n ≥ 2:

If 7n− 1 is divisible by 4 then n is an even number.

(i) Write down the contrapositive of this statement.

(ii) Proving your answer, say whether the statement is true or false. [1.5p]

Question 3

Consider the set S = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5} and define a relation R on S by (x, y) ∈ R if and only if x + 2y ≡ 0 (mod 3).

(a) Show that the relation R is symmetric.

(b) Show that the relation R is an equivalence relation and list the equivalence

class containing the element 1. [2.5p]

DMA, Mid Sweden Universityc 2 MA095G & MA098G

(3)

Question 4

Let f : R → Z be the floor function given by f (x) =b(8x + 7)/6c.

(a) Compute f(1) and f(−1).

(b) Is the function f one-to-one?

(c) Is the function f onto?

Justify your answers! [2p]

Question 5

For all integers n ≥ 2 consider the sum

sn =

n−1X

i=1

(i + 1)2i−n.

(a) Compute the sums s2, s3, s4, s5. [1p]

(b) Prove by induction that sn= n − 1 for all n ≥ 2. [2p]

Question 6

(a) How many 5-digit positive integers bigger than 63000 can be created by per-

muting the digits of the number 12369? [1p]

(b) Find the number of permutations of the digits 1, 2, 3, 4, 5, 6 which satisfy at least one of the following two conditions.

• 1 appears immediately to the left of 4

• 4 appears immediately to the left of 3 [1p]

DMA, Mid Sweden Universityc 3 MA095G & MA098G

(4)

Question 7

(a) Showing all your working, use Euclid’s algorithm to find two integers s and t such that

2014s + 3287t = gcd(2014, 3287).

[2p]

(b) Showing all your working, find all solutions x ∈ Z of the congruence 2014x ≡ 76 (mod 3287).

[1p]

Question 8

Consider the graph G on the vertex set V = {1, 2, 3, 4, 5, 6} with adjacency matrix







0 1 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0







(a) Draw G. [0.5p]

(b) Is G Hamiltonian? [0.5p]

(c) Is G bipartite? [0.5p]

(d) Is G planar? [1p]

(e) Find two non-isomorphic spanning trees of G. [1.5p]

Justify your answers!

DMA, Mid Sweden Universityc 4 MA095G & MA098G

(5)

Question 9

Describe Dijkstra’s algorithm for finding a shortest path from a vertex x to a vertex y in a weighted graph, and use it to find a shortest path from the vertex a to every

other vertex in the following graph. [2p]

gs ds

as bs

se

s i s

h

fs cs

@@

@@

@@

@@

@@

@@ 1

1

2 1

1 5 3 3

2 2

3

5 1 3

1 1

Aspect Question Let p ≥ 2 be a prime.

(a) Show for all integers n with 1 ≤ n ≤ p − 1, that the binomial coefficient npis divisible by p.

(b) Show that (x + y)p≡ xp+ yp(mod p) for all integers x and y.

DMA, Mid Sweden Universityc 5 END OF EXAMINATION

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MITTUNIVERSITETET DMA

Tentamen 2014

MA095G & MA098G Diskret matematik (svenska)

Skrivtid: 5 timmar Datum: 19 mars 2014

Pia Heidtmann

Den obligatoriska delen av denna tenta omfattar 9 fr˚agor. Delfr˚agornas po¨ang st˚ar angivna i marginalen inom [ ]-parenteser. Maximalt po¨angantal ¨ar 24.

Betyg s¨atts efter hur v¨al l¨arandem˚alen ¨ar uppfyllda. Riktv¨arde f¨or betygen ¨ar:

E: 9p D: 10p C: 14p B: 18p A: 22p

D¨arut¨over inneh˚allar skrivningen en frivillig aspektuppgift som kan h¨oja betyget om den utf¨ors v¨al med god motivering.

Behandla h¨ogst en uppgift p˚a varje papper!

Till alla uppgifter skall fullst¨andiga l¨osningar l¨amnas. Resonemang, ekvationsl¨osningar och utr¨akningar f˚ar inte vara s˚a knapph¨andiga, att de blir sv˚ara att f¨olja. Brister i framst¨allningen kan ge po¨angavdrag ¨aven om slutresultatet ¨ar r¨att!

Hj¨alpmedel: Matematisk Formelsamling Ed. 3 (delas ut), skriv- och rit- material samt minir¨aknare som ej ¨ar symbolhanterande. Ange m¨arke och modell p˚a din minir¨aknare p˚a omslaget till tentamen.

LYCKA TILL!!

DMA, Mittuniversitetetc 1 MA095G & MA098G

(7)

Uppgift 1

(a) Uttryck det hexadecimala talet 20C i basen 10. [0.5p]

(b) Ber¨akna [2]−1 [5] i Z7. [0.5p]

(c) Ange summan

s = 1· 2 · 3 + 4 · 5 · 6 + 7 · 8 · 9 + . . . + 199 · 200 · 201

m.h.a. summatecken. [1p]

(d) Ber¨akna summan

2014X

n=193

(4n − 3).

Visa dina utr¨akningar! [1p]

Uppgift 2

(a) L˚at p och q vara tv˚a logiska p˚ast˚aenden. Ge en sanningstabell f¨or f¨oljande sammansatta p˚ast˚aenden och avg¨or om de ¨ar ekvivalenta eller ej.

p1: p ⇒ q;

p2: (¬p) ⇒ (¬q). [1p]

(b) Betrakta f¨oljande p˚ast˚aende om ett heltal n ≥ 2:

Om 7n− 1 ¨ar delbart med 4 s˚a ¨ar n ett j¨amnt tal.

(i) Ange det kontrapositiva p˚ast˚aendet.

(ii) ¨Ar p˚ast˚aendet sant? Bevisa ditt svar. [1.5p]

Uppgift 3

L˚at S = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5} och definiera en relation R p˚a m¨angden S s˚adan att (x, y) ∈ R om och endast om x + 2y ≡ 0 (mod 3).

(a) Visa att relationen R ¨ar symmetrisk.

(b) Bevisa att relationen R ¨ar en ekvivalensrelation och ange ekvivalensklassen

som inneh˚aller talet 1. [2.5p]

DMA, Mittuniversitetetc 2 MA095G & MA098G

(8)

Uppgift 4

L˚at f : R → Z vara golv-funktionen

f (x) =b(8x + 7)/6c.

(a) Ber¨akna f(1) och f(−1).

(b) ¨Ar f injektiv?

(c) ¨Ar f surjektiv?

Motivera dina svar! [2p]

Uppgift 5

F¨or alla heltal n ≥ 2 definieras summan

sn =

n−1

X

i=1

(i + 1)2i−n.

(a) Ber¨akna summorna s2, s3, s4, s5. [1p]

(b) Bevisa med induktion att sn = n − 1 f¨or alla n ≥ 2. [2p]

Uppgift 6

(a) Hur m˚anga 5-siffriga positiva heltal st¨orre ¨an 63000 kan bildas genom att

permutera siffrorna i talet 12369? [1p]

(b) Best¨am antalet permutationer av siffrorna 1, 2, 3, 4, 5, 6 som uppfyller minst ett av nedanst¨aende villkor:

• ettan st˚ar omedelbart f¨ore fyran

• fyran st˚ar omedelbart f¨ore trean [1p]

DMA, Mittuniversitetetc 3 MA095G & MA098G

(9)

Uppgift 7

(a) Anv¨and Euklides algoritm f¨or att hitta tv˚a heltal s och t s˚adana att 2014s + 3287t = sgd(2014, 3287).

Visa dina utr¨akningar! [2p]

(b) Best¨am alla l¨osningar x ∈ Z till kongruensen 2014x ≡ 76 (mod 3287).

Visa dina utr¨akningar! [1p]

Uppgift 8

L˚at G vara grafen med h¨ornm¨angden V = {1, 2, 3, 4, 5, 6} och grannmatrisen







0 1 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0







(a) Rita G. [0.5p]

(b) ¨Ar G Hamiltonsk? [0.5p]

(c) ¨Ar G bipartit? [0.5p]

(d) ¨Ar G plan¨ar? [1p]

(e) Hitta tv˚a icke-isomorfa uppsp¨annande tr¨ad i G. [1.5p]

Motivera dina svar!

DMA, Mittuniversitetetc 4 MA095G & MA098G

(10)

Uppgift 9

Beskriv Dijkstras algoritm f¨or att hitta en kortaste stig fr˚an h¨orn x till h¨orn y i en viktad graf och anv¨and den f¨or att hitta en kortaste stig fr˚an h¨orn a till varje annat

h¨orn i grafen nedan. [2p]

gs ds

as bs

se

s i s

h

fs cs

@@

@@

@@

@@

@@

@@ 1

1

2 1

1 5 3 3

2 2

3

5 1 3

1 1

Aspektuppgift

L˚at p ≥ 2 vara ett primtal.

(a) Visa att p delar binomialkoefficienten np, om n ¨ar ett heltal som uppfyller att 1 ≤ n ≤ p − 1.

(b) Visa att (x + y)p≡ xp+ yp (mod p) f¨or alla heltal x och y.

DMA, Mittuniversitetetc 5 SLUT P˚A TENTAMEN

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