1 CSC 125 :: Final Exam December 14, 2011 1-5. Complete the truth tables below: p q p q p q p q p q p q T T T T F T T T F F T T F F F T F T T T F F F F F F T T (6 – 9) Let p be: Log rolling is fun. q be: The sawmill is closed. Express these as English sentences: 6. p q Log rolling is not fun or the sawmill is closed. 7. q p If log rolling is not fun then the sawmill is closed. Write these propositions using p and q and logical connectives: 8. The sawmill is not closed but logrolling is fun. q p 9. If logrolling is not fun then the sawmill is not closed. p q
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1
CSC 125 :: Final Exam December 14, 2011
1-5. Complete the truth tables below:
p q p q p q p q p q p q
T T T T F T T
T F F T T F F
F T F T T T F
F F F F F T T
(6 – 9) Let
p be: Log rolling is fun.
q be: The sawmill is closed.
Express these as English sentences:
6. p q
Log rolling is not fun or the sawmill is closed.
7. q p
If log rolling is not fun then the sawmill is closed.
Write these propositions using p and q and logical connectives:
8. The sawmill is not closed but logrolling is fun.
q p
9. If logrolling is not fun then the sawmill is not closed.
p q
2
(12 – 15) Match the following logical equivalences with the answers found in the
Answer Bank. Write the correct letter just to the right of the ≡ symbol.
Note: Some answers will be used more than once:
12. Domination Laws p T ≡ T A
p F ≡ F B
13. Identity Laws p T ≡ p C
p F ≡ p C
14. DeMorgan's Laws (p q) ≡ p q M
(p q) ≡ p q N
15. Negation Laws p p ≡ T A
p p ≡ F B
(16 – 18) Fill in the missing portion of each of the rules of inference named below:
16. Modus ponens (affirming the hypothesis) p → q
p .
q
17. Modus tollens (denying the conclusion) p → q
q .
p
18. Hypothetical syllogism p → q
q → r
p → r
(19 – 22) Determine whether the arguments below are valid or invalid, then circle
either Valid or Invalid. If valid, using the list of valid argument forms found in the
Answer Bank, write the letter of the rule of inference; if invalid write the letter of
the logical fallacy, again using the list of invalid argument forms found in the
3
Answer Bank.
19. Freedom is precious and fragile.
Freedom is fragile.
Valid Invalid
Valid – simplification F
20. If Tyler has bronchitis, then has a fever.
Tyler has a fever.
Tyler has bronchitis.
Valid Invalid
Invalid – affirming the conclusion I
21. Sam is really smart.
Sam is really smart and he gets good grades.
Valid Invalid
Invalid – unwarranted addition K
22. Logic is either hard or it is nutty.
Either logic is easy or it is impossible.
Logic is either nutty or it is impossible.
p q
p r
q r
Valid Invalid
Valid – resolution
(23 – 24) Let the domain of P(x) consist of the integers 1, 2 and 3. Write out each
proposition using disjunctions, conjunctions and negations.
23. x P(x)
P(1) P(2) P(3)
24. x P(x)
P(1) P(2) P(3)
4
(25 – 32) Let S(x) be the statement “person x stands during the 7th
inning stretch,”
where the domain for x consists of all persons who go to baseball games.
(25 – 28) Write the sentence which renders the logical expression into colloquial
English.
25. x S(x)
Everyone stands during the 7th
inning stretch
26. x S(x)
Someone stands during the 7th
inning stretch
27. x S(x)
Someone does not stand during the 7th
inning stretch
Someone sits during the 7th
inning stretch
28. x S(x)
Everyone sits during the 7th
inning stretch
No one stands during the 7th
inning stretch
(29 – 32) Render the sentences below as logical expressions, using predicates and
quantifiers.
29. Some stand during the 7th inning stretch.
x S(x)
30. Some do not stand during the 7th
inning stretch.
x S(x)
x S(x)
31. Not everyone stands during the 7th inning stretch.
x S(x)
x S(x)
32. Micah stands during the 7th
inning stretch.
S(Michah)
(33 – 35) Let F(x,y) be the statement “x can fool y,” where the domain for both x and
y consists of all people in the world. Use quantifiers to express these sentences.
33. Everyone can fool Fred.
5
xF(x,Fred)
34. Someone can fool everybody.
xyF(x,y)
35. No one can fool him- or herself.
xF(x,x) ≡ xF(x,x)
(36 – 43) Let D(x,y) be the statement “student x does homework assignment y,”
where the domain for x consists of all students in this class and the domain for y
consists of all homework assignments.
(36 – 39) Render these logical expressions into colloquial English.
36. xy D(x,y)
Someone did a homework assignment
37. x y D(x,y)
Every did every homework assignment
38. xy D(x,y)
Everyone did a homework assignment
39. yx D(x,y)
Every assignment was skipped by someone
= There is no assignment done by everyone
(40 – 43) Write the logical expression – using nested quantifiers – for each sentence
below.
40. Some students did some of the assignments.
x y D(x,y)
41. No one did every assignment.
x y D(x,y)
x y D(x,y)
42. Some did every assignment.
x yD(x,y)
43. Dana did not do any assignment.
y D(Dana,y)
6
y D(Dana,y)
(44 – 46) Match the following set identities – write the letter of the correct answer
EC-6. We know that 2/5 in base 10 is the terminating decimal 0.4, while 1/3 is the
repeating decimal 0.3333. It turns out that in base 2 the fraction ½ is the terminating
“binimal” 0.1, but 2/5 is the repeating one 0.011001100. It stands to reason that in
base 6 the fraction 1/6 terminates, since the denominator is the base itself. On the
other hand, 2/5 repeats as it did in base 2 – 0.222.
Give some other examples of fractions that terminate in base 10 but repeat in
base 6;
that terminate in base 6 but repeat in base 10;
that terminate in both base 6 and base 10;
that repeat in both base 6 and base 10.
What about fractions in base 7 vis a vis base 10?
Bases 6 & 10 – in common denominators that are powers of 2; terminate in 10 but not in 6 – denominators that are powers of 5; terminate in 6 but not in 10 – denominators that are powers of 3. Base 7 – none in common. Only those with denominators powers of 7 terminate.