COT3100 Discrete Structures and Applications 2012 fall ufl
Solutions for HW1
Problem 1
a) r ˄ ¬q
c) r→ p
e) (p ^ q) →r
Problem 2
a) Converse: “I will ski tomorrow only if it snows today.”
Contrapositive: “If I do not ski tomorrow,
then it will not have snowed today.”
Inverse: “If it does not snow today, then I will not ski tomorrow.”
b) Converse: “If I come to class, then there will be a quiz.”
Contrapositive: “If I do not come to
class, then there will not be a quiz.”
Inverse: “If there is not going to be a quiz, then I don’t come to
class.”
c) Converse: “A positive integer is a prime if it has no divisors other than 1 and itself.” Contrapositive:
“If a positive integer has a divisor other than 1 and itself, then it is not prime.”
Inverse: “If a positive
integer is not prime, then it has a divisor other than 1 and itself.”
Problem 3
Problem 4
a) Using truth table
b) Developing a series of logical equivalences
[¬p˄(p˅q)]→q
≡(¬p˄p) ˅(¬p˄q) →q
≡F˅(¬p˄q) →q
≡ (¬p˄q) →q
≡¬ (¬p˄q) ˅ q
≡ (p ˅¬ q) ˅ q
≡p ˅(¬ q˅ q)
≡p ˅T
≡T
Problem 5
a) Show (p → q) ∧ (p → r) and p → (q ∧ r) are logically equivalent.
p→(q˄r)
≡¬p˅(q˄r)
≡ (¬p v q) ˄ (¬p v r)
≡ (p → q) ∧ (p → r)
COT3100 Discrete Structures and Applications 2012 fall ufl
b) Show ￢p → (q → r) and q → (p ∨ r) are logically equivalent.
￢p → (q → r)
≡p v (q → r)
≡p v (￢q v r)
≡￢q v (p v r)
≡ q → (p ∨ r)
Problem 6
The answers given here are not unique, but care must be taken not to confuse nonequivalent sentences.
Parts (c) and (f) are equivalent; and parts (d) and (e) are equivalent. But these two pairs are not
equivalent to each other.
a) Some student in the school has visited North Dakota. (Alternatively, there exists a student in the school
who has visited North Dakota.)
b) Every student in the school has visited North Dakota. (Alternatively, all students in the school have
visited North Dakota.)
c) This is the negation of part (a): No student in the school has visited North Dakota. (Alternatively, there
does not exist a student in the school who has visited North Dakota.)
d) Some student in the school has not visited North Dakota. (Alternatively, there exists a student in the
school who has not visited North Dakota.)
e) This is the negation of part (b): It is not true that every student in the school has visited North Dakota.
(Alternatively, not all students in the school have visited North Dakota.)
Problem 7
Problem 8
Bonus Question
Alice and Charlie are lying while Bob tells the truth.
Let’s assume three propositions: A, B, C
A: Alice is telling truth;
B: Bob is telling truth;
C: Charlie is telling truth;
A must be consistent with ¬B; B must be consistent with ¬C; C must be consistent with ¬A^¬B. Only
th
the 7 situation satisfies all three requirements.
A B C ¬A ¬B ¬C ¬A^¬B
T T T F
F
F
F
F T T T
F
F
F
T F T F
T
F
F
T T F F
F
T
F
F F T T
T
F
T
T F F F
T
T
F
F T F T
F
T
F
F F F T
T
T
T