MAT 300, Mathematical Structures
Dr. L. Mantini
Problem Set 1 Solutions
Spring 2010
1.1
: 2, 4, 6, 7;
1.2
: 2, 4, 8, 12, 13, 17.
1.1.2 (3 points) Analyze the logical forms of the following statements:
(a) Either John and Bill are both telling the truth, or neither of them is.
Solution
: (J
∧
B)
∨
(
¬
J
∧ ¬
B), where
J: John is telling the truth.
B: Bill is telling the truth.
(b) I’ll have either ﬁsh or chicken, but I won’t have both ﬁsh and mashed potatoes.
Solution
: (F
∨
C)
∧ ¬
(F
∧
M), where
F: I’ll have ﬁsh.
C: I’ll have chicken.
M: I’ll have mashed potatoes.
(c) 3 is a common divisor of 6, 9, and 15.
Solution
: Here we use an abbreviation: we let TD(
n
) denote the sentence “3
divides
n
.” Then our sentence is TD(6)
∧
TD(9)
∧
TD(15).
1.1.4 Which of the following expressions are well-formed formulas?
(a)
¬
(
¬
P
∨ ¬ ¬
R): well-formed
(b)
¬
(P, Q,
∧
R): not well-formed
(c) P
∧ ¬
P, well-formed (also false—a contradiction!)
(d) (P
∧
Q)(P
∨
R): not well-formed.
1.1.6 Let S stand for the statement “Steve is happy” and G for “George is happy.” What
English sentences are represented by the following expressions?
(a) (S
∨
G)
∧
(
¬
S
∨ ¬
G): “Either Steve or George is happy, and either Steve or
George is unhappy,” or “One of Steve or George is happy, and one is unhappy.”
(b) (S
∨
[G
∧ ¬
S])
∨ ¬
G: “Steve is happy, or George is happy and Steve is unhappy,
or George is unhappy.” (Note that this sentence is a tautology.)
(c) S
∨
[G
∧
(
¬
S
∨ ¬
G)]: “Steve is happy, or George is happy and one of Steve
or George is unhappy.” This could also be said as “Steve is happy, or George is