Løsningsforslag til ukeoppgaver i MAT1030, uke 6
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1 Løsningsforslag til ukeoppgaver i MAT1030, uke 6 Sist endret: 23. februar 2010 Kap. 4 Oppgave 1 Deloppgave a) p = Karen is studying computing. q = Minh is studying mathematics. p q = Karen is studying computing and Minh is not studying mathematics. (p q) q = Either Karen is studying computing and Minh is not studying mathematics, or Minh is studying mathematics. Hovedkonnektiv (= principal connective):. Deloppgave b) p = It is sunny. q = I will carry an umbrella. p q = If it is sunny then I will carry an umbrella. (p q) = It is not the case that if it is sunny then I will carry an umbrella. Hovedkonnektiv:. 1
2 Deloppgave c) p = The program will terminate. q = The input is numeric. r = The escape key is pressed. q r = The input is not numeric or the escape key is pressed. p ( q r) = The program will terminate if and only if the input is not numeric or the escape key is pressed. Hovedkonnektiv:. Deloppgave d) p = x = 7. q = y = 4. r = z = 2. p q r = x = 7 and y 4 and z = 2. (q r) = It is not true that either y = 4 or z 2. (q r) (p r) = If it is not true that either y = 4 or z 2 then x = 7 or z = 2. (p q r) ( (q r) (p r)) = If x = 7 and y 4 and z = 2, then if it is not true that either y = 4 or z 2 then x = 7 or z = 2. 2
3 Oppgave 2 p = It is snowing. q = I will go skiing. Deloppgave a) p q = It is not snowing and I will go skiiing. Deloppgave b) p q = If it is snowing then I will go skiing. Deloppgave c) q p = If I will not go skiing then it is snowing. Deloppgave d) p q = It is snowing or I will not go skiing. (p q = It is snowing or I will not go skiing, and it is snowing. 3
4 Oppgave 3 Dette var vi innom i forrige uke der vi brukte symbolet i stedet for. Deloppgave a) p q p q T T F T F T F T T F F F Deloppgave b) p q p q (p q) (p q) (p q) p q T T T F F F T F T T T T F T T T T T F F F T F F Altså, p q (p q) (p q). 4
5 Oppgave 4 Deloppgave a) p = The input file exists. q = An error message is generated. p q = If the input file exists, then an error message is not generated. Omvendt: p q q p = If an error message is not generated, then the input file exists. Kontrapositivt: (p q) p q q p = If an error message is generated, then the input file does not exists. Deloppgave b) p = The database is accessible. q = My program can run. p q = If the database is not accessible, then my program cannot run. Omvendt: p q q p = If my program cannot run, then the database is not accessible. Kontrapositivt: ( p q) p q q p = If my program can run, then the database is accessible. Deloppgave c) p = My program contains bugs. q = It produces correct output. p q = If my program contains no bugs, then it produces correct output. Omvendt: p q q p = If it produces correct output, then my program contains no bugs. Kontrapositivt: ( p q) p q q p = If it does not produce correct output, then my program contains bugs. 5
6 Oppgave 6 Deloppgave a) p q p q (p q) p T T T T T F T T F T F T F F T F Ingen av delene. Deloppgave b) p q p (p q) (p (p q)) q T T T F T F F T F T T F F F T T Ingen av delene. Deloppgave c) p q p q p q (p q) ( p q) T T T F F T F F T F F T F T F F F F T F Dette er en selvmotsigelse (kontradiksjon). 6
7 Deloppgave d) p q r p r q r p q (p r) (q r) ((p r) (q r)) (p q) T T T T T F T F T T F F F F F T T F T T F T T T T F F F F T F T F T T F T T T T F T F F F T F T F F T F F T F T F F F F F T F T Ingen av delene. 7
8 Oppgave 7 Deloppgave a) Hvis P er sann så er P Q sann uavhengig av hva Q er. Deloppgave b) i) Her er det som er det ytterste konnektivet, så påstanden er usann hvis (p r) q eller (p q) (p r) er usann. (p q) (p r) er usann hvis (p q) eller p r er usann. Tilsvarende er påstanden sann hvis (p q), p r og (p r) q alle er sanne. p q r (p q) p r (p r) q ( (p q) (p r)) ((p r) q) T T T F F T T F F F T F T T T T T T F F T T T T F T T F F F T F F F F F T F F F F F T T T T ii) Her er det som er det ytterste konnektivet, så påstanden er sann hvis ( p (q r)) eller p r er sann. Tilsvarende er påstanden usann hvis ( p (q r)) og p r er usanne. p q r ( p (q r)) p r ( p (q r)) ( p r) T T T T T T T F T T T F T T T T F F T T F T T F F F F T F T T F F T F F F F F F T T 8
9 Oppgave 9 Deloppgave a) p q r q r p q p r p (q r) (p q) (p r) T T T T T T T T T T F T T F T T T F T T F T T T T F F F F F F F F T T T F F F F F T F T F F F F F F T T F F F F F F F F F F F F Altså, er distributiv over : Deloppgave b) p (q r) (p q) (p r). p q p q p (p q) T T T T T F T T F T T F F F F F 9
10 Oppgave 11 p = x 3. q = x < 6. not(x 3 and x < 6) = (p q) = p q = x < 3 or x 6. Så en enklere måte å skrive dette på er følgende, If x < 3 or x 6 then... 10
11 Oppgave n 0 2. term 1 3. sum 0 4. n n term term/2 6. sum sum + term 7. While term and n 100 do 7.1 n n term term/2 7.3 sum sum + term 11
12 Oppgave 16 p q ( p q) p q p q ( p q) T T T T T F T T F T T T F F F F 12
13 Oppgave 17 p q p q (p p) p (p q) (p q) p q (p p) (q q) p q T T F F T T T F T F F T F T T T F T F F T T F F 13
14 Andre oppgaver Oppgave fra forelesning p q p q q p (p q) p (p q) (q p) T T T T T T T F F T T T F T T F T T F F T T T T 14
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