Name___________________________________ Spring 2009
MAT101-Exam 2 Supplemtal Review Prof. T. Koukounas
Decide whether or not the following is a statement.
1) My favorite baseball team will win the pennant.
2) Do you like this color?
Decide whether the statement is compound.
3) 5 is rational and 6 is irrational.
4) Today is not Thursday.
5) Computers are very helpful to people.
Write a negation for the statement.
6) She earns more than me.
7) Everyone is asleep.
8) No fifth graders play soccer.
Write a negation of the inequality. Do not use a slash symbol.
9) x ≥ -54
10) x < 94
Convert the symbolic compound statement into words.
11) p represents the statement ʺItʹs Monday.ʺ
q represents the statement ʺItʹs raining today.ʺ
Translate the following compound statement into words:
~p ∨ ~q
Let p represent the statement, ʺJim plays footballʺ, and let q represent the statement ʺMichael plays basketballʺ. Convert
the compound statement into symbols.
12) Jim does not play football and Michael does not play basketball.
13) Jim does not play football and Michael plays basketball.
Decide whether the statement is true or false.
14) Every rational number is an integer.
15) All whole numbers are real numbers.
16) There exists a rational number that is an integer.
17) Some real numbers are integers.
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18) At least one irrational number is not an integer.
19) No rational number is not a whole number.
Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound
statement.
20) p ∧ (q ∨ p)
21) p ∨ ~q
22) ~[(~p ∧ ~q) ∨ ~q]
Let p represent a true statement, while q and r represent false statements. Find the truth value of the compound
statement.
23) ~[(~p ∧ q) ∨ r]
24) (p ∧ ~q) ∧ r
Let p represent 7 < 8, q represent 2 < 5 < 6, and r represent 3 < 2. Decide whether the statement is true or false.
25) ~p ∨ q
26) ~(~p ∧ ~q) ∧ (~r ∧ ~q)
Give the number of rows in the truth table for the compound statement.
27) p ∧ (~q ∧ r)
28) ~(p ∨ q) ∧ (w ∧ ~s) ∨ (r ∨ t) ∧ (~u ∧ s)
Construct a truth table for the statement.
29) ~r ∧ ~s
30) (p ∧ ~s) ∧ q
Use De Morganʹs laws to write the negation of the statement.
31) 6 < 8 or 11 ≠ 13
32) 8 + 4 = 12 and 10 - 2 ≠ 8
Decide whether the statement is true or false.
33) For every real number x, x < 5 and x > 4.
34) For no real number y, y < 17 and y > 19.
35) For every real number r, r < 8 or r > 7.
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Solve the problem.
36) Given that p ∧ q is true, what can you conclude about the truth values of p and q?
A) Both p and q are true
B) Exactly one of p and q is true
C) At least one of p and q must be true
D) p and q have the same truth value
37) Given that ~(p ∧ q) is true, what can you conclude about the truth values of p and q?
A) Both p and q are false
B) Exactly one of p and q is true
C) Both p and q are true
D) At least one of p and q is false
38) Given that p ∨ q is false, what can you conclude about the truth values of p and q?
A) p and q have the same truth value
B) Exactly one of p and q is false
C) Both p and q are false
D) At least one of p and q is false
39) Given that (p ∨ q) ∨ ~p is true, what can you conclude about the truth values of p and q?
A) q is true
B) p is true
C) At least one of p and q is true
D) p and q can have any truth values
40) Given that (p ∧ q) ∨ ~q is false, what can you conclude about the truth values of p and q?
A) p and q can have any truth values
B) p is false, q can be either true or false
C) At least one of p and q is true
D) p is false and q is true
Rewrite the statement using the if...then connective. Rearrange the wording or words as necessary.
41) Cats chase mice.
42) All children like stuffed toys.
Tell whether the conditional statement is true or false.
43) Here T represents a true statement.
T → (2 = 7)
44) Here T represents a true statement.
T → (5 < 3)
45) Here F represents a false statement.
(2 = 2) → F
Write the compound statement in words.
Let r = ʺThe puppy is trained.ʺ
p = ʺThe puppy behaves well.ʺ
q = ʺHis owners are happy.ʺ
46) ~(p → q)
Write the compound statement in symbols.
Let r = ʺThe food is good.ʺ
p = ʺI eat too much.ʺ
q = ʺIʹll exercise.ʺ
47) If I exercise, then I wonʹt eat too much.
48) The food is good and if I eat too much, then Iʹll exercise.
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Given p is true, q is true, and r is false, find the truth value of the statement.
49) ~q → r
50) ~r → ~p
51) ~q → (p ∨ r)
52) ~[(~q → r) →(q ∨ r)]
53) [(~p → r) ∧ (~p ∨ q)] → r
Construct a truth table for the statement.
54) (q → ~p) → (q ∧ ~p)
Tell whether the conditional statement is true or false.
55) Here F represents a false statement.
(9 < 5) → F
Write the negation of the conditional. Use the fact that the negation of p → q is p ∧ ~q.
56) If you give your rain coat to the doorman, he will give you a dirty look.
True or false?
57) When using a truth table, the statement ~(q → p) is equivalent to q ∧ ~p.
58) When using a truth table, the statement q → p is equivalent to p → q.
Write a logical statement representing the following circuit. Simplify when possible.
59)
60)
Draw a circuit representing the following statement as it is given. Simplify if possible.
61) ~p → [(q ∧ r) ∨ ~p]
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62) p ∧ [(r ∨ q) ∨ ~p]
Decide whether the statement is true or false.
63) If p is false then the statement q → (~p ∨ q) must be true.
64) If a conditional statement is false, its consequent must be false.
65) If q is false then the statement (p ∧ q) → p must be true.
66) If a conditional statement is true, its consequent must be true.
Write the converse, inverse, or contrapositive of the statement as requested.
67) If I pass, Iʹll party.
Contrapositive
68) All Border Collies are dogs.
Inverse
Rewrite the statement in the form ʺif p, then qʺ.
69) I will lose weight if I diet.
70) Practice is necessary for making the team.
Find the truth value of the statement.
71) 7 - 5 = 2 if and only if 8 + 5 = 14.
72) 4 + 7 ≠ 13 if and only if 8 × 5 ≠ 45.
Use an Euler diagram to determine whether the argument is valid or invalid.
73) All cats like fish.
Henry does not like fish.
Henry is not a cat.
74) All businessmen wear suits.
Aaron wears a suit. Aaron is a businessman.
75) Some cars are considered sporty.
Some cars are safe at high speeds.
Some sports cars are safe at high speeds.
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Answer Key
Testname: MAT101_EX2REV_SP09
1) Not a statement
2) Not a statement
3) Compound
4) Compound
5) Not compound
6) She does not earn more than me.
7) Not everyone is asleep.
8) At least one fifth grader plays soccer.
9) x < -54
10) x ≥ 94
11) Itʹs not Monday or itʹs not raining today.
12) ~p ∧ ~q
13) ~p ∧ q
14) False
15) True
16) True
17) True
18) True
19) False
20) True
21) True
22) False
23) True
24) False
25) True
26) False
27) 8
28) 128
29) r s (~r ∧ ~s)
T
T
F
F
30) p
T
F
T
F
s
q
F
F
F
T
(p ∧ ~s) ∧ q
T T T
F
T T F
F
T F T
T
T F F
F
F T T
F
F T F
F
F F T
F
F F F
F
31) 6 ≥ 8 and 11 = 13
32) 8 + 4 ≠ 12 or 10 - 2 = 8
33) FALSE
34) TRUE
35) TRUE
36) A
37) D
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Answer Key
Testname: MAT101_EX2REV_SP09
38) C
39) D
40) D
41) If it is a cat, then it chases mice.
42) If it is a child, then it likes stuffed toys.
43) False
44) False
45) False
46) It is not the case that if the puppy behaves well then his owners are happy.
47) q → ~p
48) r ∧ (p → q)
49) True
50) False
51) True
52) False
53) False
54) q p (q → ~p) → (q ∧ ~p)
T T
T
T F
T
F T
F
F F
F
55) True
56) You give your rain coat to the doorman and he will not give you a dirty look.
57) True
58) False
59) p ∧ (r ∨ q)
60) (p ∧ ~q) ∨ (~p ∧ q); The statement simplifies to q.
61) The statement simplifies to T.
62) The statement simplifies to p ∧ (r ∨ q).
63) True
64) True
65) True
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Answer Key
Testname: MAT101_EX2REV_SP09
66) False
67) If I donʹt party, I didnʹt pass.
68) If itʹs not a Border Collie, itʹs not a dog.
69) If I diet, then Iʹll lose weight.
70) If you make the team, then you must have practiced.
71) False
72) True
73) Valid
74) Invalid
75) Invalid
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