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Discrete Mathematics Exam - Questions and Answers

Instructions

There are 30 questions in total, make sure you attempt all and number them correctly on this document and in your working notes.

Total = 100 Marks

1. Suppose p is true, q is true, s is false.

Find the truth value of (~s?p) ?Â (q?~s) (4 MARKS)

2. Suppose p is false, s is false, r is true.

Find the truth value of ~[(s?p) ?Â ~r] (4 MARKS)

3. Suppose p is false, q is true, s is true.

Find the truth value of (p?~q) ?Â ~s (4 MARKS)

4. Suppose p is false, q is true, r is false.

Find the truth value of (p?~q) ?Â r (4 MARKS)

5. Suppose p is true, q is true, r is true, s is false.

Find the truth value of (~p?s) ?Â (s?r) (4 MARKS)

6. ~[(p?~q) ?~p]

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7. (p?q) ? ~(~q?r) Â

8. True or false: (~p?q) ?(p?~q) is a tautology.

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9. True or false: (~p?q) ?(~p?q)? (~p?q)

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10. True or false: (p?q) ?~(~q?r) ?(p?~q) ? r Â Â

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11. Select the statement that is the negation of (3 MARKS)

â€œCoach Spurrier is charming and Coach Spurrier is modest.â€

a. Coach Spurrier is not charming and Coach Spurrier is not modest.

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b. Coach Spurrier is not charming or Coach Spurrier is not modest.

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c. Coach Spurrier is not charming and Coach Spurrier is modest.

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d. Letâ€™s get serious for a minute.

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12. Select the statement that is the negation of Â (3 MARKS)

â€œThe speed limit is 55 and granny is driving 35.â€

a. The speed limit is not 55 or granny is not driving 35.

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b. The speed limit is not 55 and granny is not driving 35

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c. The speed limit is not 55 or granny is driving 35.

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d. The speed limit is not 55 and granny is driving 35.

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e. Counseling for road rage is available at 1 - 900 - calmdown.

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13. Select the statement that is the negation of

â€œAll circus clowns are undignified.â€ (3 MARKS)

a. All circus clowns are dignified. Â

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b. All cirrus clouds are indistinguishable.

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c. Some circus clowns are dignified.

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d. No circus clowns are dignified.

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14. Suppose the marked diagram below conveys information about relationships between pirates, ruffians and scoundrels. We use shading to indicate that a region contains no elements. An â€œXâ€ in a region indicates the existence of at least one element; an â€œXâ€ on the boundary between two regions indicates that the union of those two regions contains at least one element. If a region is unmarked, then whether that region contains any elements is uncertain. Select the choice that must be true according to the marked diagram. (3 MARKS)

Â a. No scoundrels are ruffians and some pirates arenâ€™t scoundrels.

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b. All ruffians are scoundrels and some scoundrels arenâ€™t pirates.

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c. No pirates are scoundrels and some ruffians are scoundrels.

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d. All ruffians are scoundrels and some pirates are ruffians.

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e. None of these is correct.

15. 11101+10011+110111

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16. 111+101+100

In a certain school, there are 180 pupils in Year 7. One hundred and ten pupils study French, 88 study German and 65 study Indonesian. Forty pupils study both French and German, 38 study German and German only. Find the number of pupils who study: Show in Venn Diagram and write correct answers. (10 MARKS)Â

17. all three languages

18. Indonesian only

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19. none of the languages

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20. at least one language

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21. either one or two of the three languages

22. A = { whole numbers less than 100 divisible by 16 }.

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23. Specify the set B by giving a written description of its elements, where B = { 0, 1, 4, 9, 16, 25 }.

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24. Does the following sentence specify a set? C = { whole numbers close to 50 }.

25. How many people enjoy all three sports?

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26. How many people enjoy football and hockey but not rugby?

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27. How many people enjoy football and rugby but not hockey?

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28. Work out which sport is enjoyed by the most number of people.

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29. Let A = {a, e, l, t, r}. Which of the following sets are equal to A? (Choose all that apply.) and show working. (6 MARKS)

(a) {x | x is a letter of the word latter}

(b) {x | x is a letter of the word later}

(c) {x | x is a letter of the word late}

(d) {x | x is a letter of the word rated}

(e) {x | x is a letter of the word relate}

30. If : Â (6 MARKS)

Find: (AUB')'