1. Test #2 Practice
MGF 1106 Summer 2011

2. 3.1 Statements and Logical Connectives

3. Question #1
Indicate whether the statement is a simple or a compound statement. If it is a compound statement, indicate whether it is a negation, conjunction, disjunction, conditional, or biconditional by using both the word and its appropriate symbol.
It is false that whales are fish and bats are birds.
Compound; negation; ~
Compound; disjunction; V
Compound; conjunction; 
Compound; biconditional; 

4. Question #1
Indicate whether the statement is a simple or a compound statement. If it is a compound statement, indicate whether it is a negation, conjunction, disjunction, conditional, or biconditional by using both the word and its appropriate symbol.
It is false that whales are fish and bats are birds.
Compound; negation; ~
Compound; disjunction; V
Compound; conjunction; 
Compound; biconditional; 

5. Question #2 Write a negation of the statement.
All dinosaurs were carnivores.
Some dinosaurs were carnivores.
All dinosaurs were not carnivores.
Some dinosaurs were not carnivores.
No dinosaurs were carnivores.

6. Question #2 Write a negation of the statement.
All dinosaurs were carnivores.
Some dinosaurs were carnivores.
All dinosaurs were not carnivores.
Some dinosaurs were not carnivores.
No dinosaurs were carnivores.

7. Question #3
Let p represent the statement, “Jim plays football”, and let q represent “Michael plays basketball”. Convert the compound statements into symbols.
It is not the case that Jim does not play football and Michael does not play basketball.
~(p V q)
~p  ~q
~(~p V ~q)
~(~p  ~q)

8. Question #3
Let p represent the statement, “Jim plays football”, and let q represent “Michael plays basketball”. Convert the compound statements into symbols.
It is not the case that Jim does not play football and Michael does not play basketball.
~(p V q)
~p  ~q
~(~p V ~q)
~(~p  ~q)

9. Question #4 Write the compound statement in symbols.
r = “The food is good,”
p = “I eat too much,”
q = “I’ll exercise.”
I’ll exercise if I eat too much.
q  p
p V q
p  q
q  p

10. Question #4 Write the compound statement in symbols.
r = “The food is good,”
p = “I eat too much,”
q = “I’ll exercise.”
I’ll exercise if I eat too much.
q  p
p V q
p  q
q  p

11. Question #5 Convert the compound statement into words.
p = “Babies eat bananas.”
q = “Babies wear plastic bibs.”
p ∨ q
Babies eat bananas or babies do not wear plastic bibs.
Babies eat bananas or babies wear plastic bibs.
Babies wear plastic bibs and babies eat bananas.
Babies eat bananas and babies wear plastic bibs.

12. Question #5 Convert the compound statement into words.
p = “Babies eat bananas.”
q = “Babies wear plastic bibs.”
p ∨ q
Babies eat bananas or babies do not wear plastic bibs.
Babies eat bananas or babies wear plastic bibs.
Babies wear plastic bibs and babies eat bananas.
Babies eat bananas and babies wear plastic bibs.

13. Question #6 Write the compound statement in words.
r = “The puppy is trained.”
p = “The puppy behaves well.”
q = “His owners are happy.”
r ∧ (p → q)
The puppy is trained if the puppy behaves well and his owners are happy.
The puppy is trained, or if the puppy behaves well then his owners are happy.
If the puppy is trained then the puppy behaves well, and his owners are happy.
The puppy is trained, and if the puppy behaves well then his owners are happy.

14. Question #6 Write the compound statement in words.
r = “The puppy is trained.”
p = “The puppy behaves well.”
q = “His owners are happy.”
r ∧ (p → q)
The puppy is trained if the puppy behaves well and his owners are happy.
The puppy is trained, or if the puppy behaves well then his owners are happy.
If the puppy is trained then the puppy behaves well, and his owners are happy.
The puppy is trained, and if the puppy behaves well then his owners are happy.

15. Question #7
A restaurant has the following statement on the menu: “All dinners are served with a choice of: Soup or Salad, and Potatoes or Pasta, and Corn or Beans.” A customer asks for soup, salad, and corn. Is this order permissible?
Yes No

16. Question #7
A restaurant has the following statement on the menu: “All dinners are served with a choice of: Soup or Salad, and Potatoes or Pasta, and Corn or Beans.” A customer asks for soup, salad, and corn. Is this order permissible?
Yes No

17. Question #8
Select letters to represent the simple statements and write each statement symbolically by using parentheses then indicate whether the statement is a negation, conjunction, disjunction, conditional, or biconditional.
If a number is divisible by 3 and the number is not divisible by 2 then the number is not divisible by 6.
p ∧ (~q → ~r); conjunction
(p ∧ ~q) → (~r); conditional
(p ∧ ~q) ↔ (~r); biconditional
p ∨ (~q → ~r); disjunction

18. Question #8
Select letters to represent the simple statements and write each statement symbolically by using parentheses then indicate whether the statement is a negation, conjunction, disjunction, conditional, or biconditional.
If a number is divisible by 3 and the number is not divisible by 2 then the number is not divisible by 6.
p ∧ (~q → ~r); conjunction
(p ∧ ~q) → (~r); conditional
(p ∧ ~q) ↔ (~r); biconditional
p ∨ (~q → ~r); disjunction

19. 3.2 Truth Tables for Negation, Conjunction, and Disjunction

20. Question #9
Determine how many distinct cases must be listed in the truth table for the compound statement.
(p ∨ q) ∨ (~r ∨ s) ∧ ~t 8 32 10 25
(p ∨ q) ∨ (~r ∨ s) ∧ ~t

21. Question #9
Determine how many distinct cases must be listed in the truth table for the compound statement.
(p ∨ q) ∨ (~r ∨ s) ∧ ~t 8 32 10 25
(p ∨ q) ∨ (~r ∨ s) ∧ ~t

22. Question #10 Construct a truth table for the statement.
(s ∧ q) ∧ (~q ∨ t) a) b) s q t
(s  q) 
(~q V t) T F s q t
(s  q) 
(~q V t) T F

23. Question #10 Construct a truth table for the statement.
(s ∧ q) ∧ (~q ∨ t) a) b) s q t
(s  q) 
(~q V t) T F s q t
(s  q) 
(~q V t) T F

24. Question #11
Translate the statement into symbols then construct a truth table. p = At most, 100 guests arrived at the wedding reception. q = There was a lot of cake left over. It is not the case that, at most, 100 guests arrived at the wedding reception and there was a lot of cake left over.
a) b) c) d) p q
~(p  q) T F p q
~(p  q) T F p q
~(p  q) T F p q
~(p  q) T F

25. Question #11
Translate the statement into symbols then construct a truth table. p = At most, 100 guests arrived at the wedding reception. q = There was a lot of cake left over. It is not the case that, at most, 100 guests arrived at the wedding reception and there was a lot of cake left over.
a) b) c) d) p q
~(p  q) T F p q
~(p  q) T F p q
~(p  q) T F p q
~(p  q) T F

26. Question #12
Let p represent a true statement, while q and r represent false statements. Find the truth value of the compound statement.
~[~p ∧ (~q ∨ p)]
true
false

27. Question #12
Let p represent a true statement, while q and r represent false statements. Find the truth value of the compound statement.
~[~p ∧ (~q ∨ p)]
true
false

28. Question #13
Determine the truth value for each simple statement. Then use these truth values to determine the truth value of the compound statement. Use a reference source such as an almanac or the chart or graph provided.
Ronald Reagan was President of the United States or Marie Curie won the Nobel Prize, and Michael Jordan played soccer.
true
false

29. Question #13
Determine the truth value for each simple statement. Then use these truth values to determine the truth value of the compound statement. Use a reference source such as an almanac or the chart or graph provided.
Ronald Reagan was President of the United States or Marie Curie won the Nobel Prize, and Michael Jordan played soccer.
true
false

30. 3.3 Truth Tables for the Conditional and Biconditional

31. Question #14 Construct a truth table for the statement.
(~p → q) ↔ (q → ~r) a) b) p q r
(~pq) 
(q~r) T F p q r
(~pq) 
(q~r) T F

32. Question #14 Construct a truth table for the statement.
(~p → q) ↔ (q → ~r) a) b) p q r
(~pq) 
(q~r) T F p q r
(~pq) 
(q~r) T F

33. Question #15
Write the compound statement in symbols. Then construct a truth table for the symbolic statement.
Let r = “The food is good,” p = “I eat too much,”
q = “I’ll exercise.”
The food is good and if I eat too much, then I’ll exercise.
a) r ∧ (p → q)
b) (r ∧ p) → q
c) (r ∧ p) → q p q r 
(pq) T F p q r 
(pq) T F p q r 
(pq) T F

34. Question #15
Write the compound statement in symbols. Then construct a truth table for the symbolic statement.
Let r = “The food is good,” p = “I eat too much,”
q = “I’ll exercise.”
The food is good and if I eat too much, then I’ll exercise.
a) r ∧ (p → q)
b) (r ∧ p) → q
c) (r ∧ p) → q p q r 
(pq) T F p q r 
(pq) T F p q r 
(pq) T F

35. Question #16
Determine whether the statement is a self-contradiction, an implication, a tautology (that is not also an implication), or none of these.
[(p → q) ∧ (q → r)] → (p → r)
Self-contradiction
Implication
Tautology
None of these

36. Question #16
Determine whether the statement is a self-contradiction, an implication, a tautology (that is not also an implication), or none of these.
[(p → q) ∧ (q → r)] → (p → r)
Self-contradiction
Implication
Tautology
None of these

37. Question #17
Given p is true, q is true, and r is false, find the truth value of the statement.
~[(~q → r) ↔(q ∨ r)]
False
True

38. Question #17
Given p is true, q is true, and r is false, find the truth value of the statement.
~[(~q → r) ↔(q ∨ r)]
False
True

39. Question #18
Determine the truth value for each simple statement. Then, using the truth values, give the truth value of the compound statement.
8 × 2 = 20 if and only if = 15.
True
False

40. Question #18
Determine the truth value for each simple statement. Then, using the truth values, give the truth value of the compound statement.
8 × 2 = 20 if and only if = 15.
True
False

41. Question #19
Use the information given in the chart or graph to determine the truth values of the simple statements. Then determine the truth value of the compound statement given. Moon 2 has a diameter of 5-9 km and Moon 4 may have an atmosphere, or Moon 5 does not have a diameter of km. a) False b) True
Planet X
 Moon 1
Diameter of Moons:
May have:
 Moon 2
 5-9 km
 Water ice
 Moon 3
 km
 atmosphere
 Moon 4
 km
 both
 Moon 5

42. Question #19
Use the information given in the chart or graph to determine the truth values of the simple statements. Then determine the truth value of the compound statement given. Moon 2 has a diameter of 5-9 km and Moon 4 may have an atmosphere, or Moon 5 does not have a diameter of km. a) False b) True
Planet X
 Moon 1
Diameter of Moons:
May have:
 Moon 2
 5-9 km
 Water ice
 Moon 3
 km
 atmosphere
 Moon 4
 km
 both
 Moon 5

43. 3.4 Equivalent Statements

44. Question #20
Use DeMorgan’s laws or a truth table to determine whether the two statements are equivalent.
~(p ∨ q) , ~p ∧ ~q
Equivalent
Not equivalent

45. Question #20
Use DeMorgan’s laws or a truth table to determine whether the two statements are equivalent.
~(p ∨ q) , ~p ∧ ~q
Equivalent
Not equivalent

46. Question #21 Write an equivalent sentence for the statement.
If it is raining, you take your coat. (Hint: Use the fact that p → q is equivalent to ~p ∨ q.)
It is not raining and you take your coat.
It is not raining and you do not take your coat.
If it is raining, you do not take your coat.
It is not raining or you take your coat.

47. Question #21 Write an equivalent sentence for the statement.
If it is raining, you take your coat. (Hint: Use the fact that p → q is equivalent to ~p ∨ q.)
It is not raining and you take your coat.
It is not raining and you do not take your coat.
If it is raining, you do not take your coat.
It is not raining or you take your coat.

48. Question #22
Write the indicated statement. Use De Morgan’s Laws if necessary.
If the moon is out, then we will start a campfire and we will roast marshmallows.
Inverse
If we do not start a campfire or we do not roast marshmallows, then the moon is not out.
If the moon is not out, then we will start a campfire but we will not roast marshmallows.
If we start a campfire and we roast marshmallows, then the moon is out.
If the moon is not out, then we will not start a campfire or we will not roast marshmallows.
If the moon is not out, then we will not start a campfire or we will not roast marshmallows.

49. Question #22
Write the indicated statement. Use De Morgan’s Laws if necessary.
If the moon is out, then we will start a campfire and we will roast marshmallows.
Inverse
If we do not start a campfire or we do not roast marshmallows, then the moon is not out.
If the moon is not out, then we will start a campfire but we will not roast marshmallows.
If we start a campfire and we roast marshmallows, then the moon is out.
If the moon is not out, then we will not start a campfire or we will not roast marshmallows.
If the moon is not out, then we will not start a campfire or we will not roast marshmallows.

50. Question #23
Write the contrapositive of the statement. Then use the contrapositive to determine whether the conditional statement is true or false.
If 1/n is not an integer, then n is not an integer.
If n is an integer, then 1/n is an integer. false
If 1/n is an integer, then n is an integer. true
If n is an integer, then 1/n is an integer. true
If 1/n is an integer, then n is an integer. false

51. Question #23
Write the contrapositive of the statement. Then use the contrapositive to determine whether the conditional statement is true or false.
If 1/n is not an integer, then n is not an integer.
If n is an integer, then 1/n is an integer. false
If 1/n is an integer, then n is an integer. true
If n is an integer, then 1/n is an integer. true
If 1/n is an integer, then n is an integer. false

52. Question #24 I and II are equivalent II and III are equivalent
Determine which, if any, of the three statements are equivalent.
I) If the garden needs watering, then the garden needs weeding or the garden is not lovely.
II) The garden needs watering, and it is false that the garden does not need weeding and the garden is not lovely.
III) The garden needs watering, and the garden needs weeding or the garden is lovely.
I and II are equivalent
II and III are equivalent
I and III are equivalent
None are equivalent

53. Question #24 I and II are equivalent II and III are equivalent
Determine which, if any, of the three statements are equivalent.
I) If the garden needs watering, then the garden needs weeding or the garden is not lovely.
II) The garden needs watering, and it is false that the garden does not need weeding and the garden is not lovely.
III) The garden needs watering, and the garden needs weeding or the garden is lovely.
I and II are equivalent
II and III are equivalent
I and III are equivalent
None are equivalent

54. 3.5 Symbolic Arguments

55. Question #25 Use truth tables to test the validity of the argument.
p ∨ q q
∴ p
valid
invalid

56. Question #25 Use truth tables to test the validity of the argument.
p ∨ q q
∴ p
valid
invalid

57. Question #26
Determine if the argument is valid or invalid. Give a reason to justify answer.
If you wear a tie, then you look natty.
You do not look natty.
∴ You are not wearing a tie.
Valid by the law of contraposition
Invalid by fallacy of the inverse
Valid by disjunctive syllogism
Invalid by fallacy of the converse

58. Question #26
Determine if the argument is valid or invalid. Give a reason to justify answer.
If you wear a tie, then you look natty.
You do not look natty.
∴ You are not wearing a tie.
Valid by the law of contraposition
Invalid by fallacy of the inverse
Valid by disjunctive syllogism
Invalid by fallacy of the converse

59. Question #27 Determine whether the argument is valid or invalid.
Sam plays tennis or George is not a man. If George is not a man, then Betsy does not win the award. Betsy wins the award. Therefore, Sam does not play tennis.
Invalid
Valid

60. Question #27 Determine whether the argument is valid or invalid.
Sam plays tennis or George is not a man. If George is not a man, then Betsy does not win the award. Betsy wins the award. Therefore, Sam does not play tennis.
Invalid
Valid

61. Question #28
Use the method of writing each premise in symbols in order to arrive at a valid conclusion.
Smiling people are happy.
Alert people are not happy.
Careful drivers are alert.
Careless drivers have accidents.
Therefore, ...
Careful drivers have accidents.
Careful drivers are happy.
People who smile are alert.
People who smile have accidents.

62. Question #28
Use the method of writing each premise in symbols in order to arrive at a valid conclusion.
Smiling people are happy.
Alert people are not happy.
Careful drivers are alert.
Careless drivers have accidents.
Therefore, ...
Careful drivers have accidents.
Careful drivers are happy.
People who smile are alert.
People who smile have accidents.

63. 3.6 Euler Diagrams and Syllogistic Arguments

64. Question #29
Use an Euler diagram to determine whether the syllogism is valid or invalid.
All businessmen wear suits.
Aaron wears a suit.
∴ Aaron is a businessman.
Invalid
Valid

65. Question #29
Use an Euler diagram to determine whether the syllogism is valid or invalid.
All businessmen wear suits.
Aaron wears a suit.
∴ Aaron is a businessman.
Invalid
Valid