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Math 310 1–28–
Solutions to Problem Set 2
Exercise 2.16. Consder the statements
P:
2 is rational
Q:
is rational.
Write each of the following statements in words and indicate whether it is true or false.
First, note that P is false and Q is true.
(a) P → Q.
The statement is “If
2 is rational, then
is rational”. Since “
2 is rational” is false, the statement
is true.
(b) Q → P.
The statement is “If
is rational, then
2 is rational”. Since “
is rational” is true and “
2 is
rational” is false, the statement is false.
(c) ∼ P →∼ Q.
The statement is “If
2 isn’t rational, then
isn’t rational”. Since “
2 isn’t rational” is true and
isn’t rational” is false, the statement is false.
(d) ∼ Q →∼ P.
The statement is “If
isn’t rational, then
2 isn’t rational”. Since “
isn’t rational” is false, the
statement is true.
Exercise 2.62. (a) For statements P , Q, and R, show that
((P ∧ Q) → R) ↔ ((P ∧ ∼ R) →∼ Q).
P Q R P ∧ Q (P ∧ Q) → R ∼ Q ∼ R P ∧ ∼ R (P ∧ ∼ R) →∼ Q
T T T T T F F F T
T T F T F F T T F
T F T F T T F F T
T F F F T T T T T
F T T F T F F F T
F T F F T F T F T
F F T F T T F F T
F F F F T T T F T
Since the columns for (P ∧ Q) → R and (P ∧ ∼ R) →∼ Q are the same, the statements are logically
equivalent.
(b) For statements P , Q, and R, show that
((P ∧ Q) → R) ↔ ((Q∧ ∼ R) →∼ P ).
P Q R P ∧ Q (P ∧ Q) → R ∼ P ∼ R Q∧ ∼ R (Q∧ ∼ R) →∼ P
T T T T T F F F T
T T F T F F T T F
T F T F T F F F T
T F F F T F T F T
F T T F T T F F T
F T F F T T T T T
F F T F T T F F T
F F F F T T T F T
Since the columns for (P ∧ Q) → R and (Q∧ ∼ R) →∼ P are the same, the statements are logically
equivalent.
- Suppose that
“Calvin likes pepperoni pizza” is true.
“Phoebe likes chocolate milkshakes” is true.
“Bonzo likes curly fries” is false.
Use a one-line truth table to determine the truth or falsity of: “If either Calvin doesn’t like pepperoni
pizza or Phoebe likes chocolate milkshakes, then either Phoebe doesn’t like chocolate milkshakes or Bonzo
likes curly fries.”
Let
P = “Calvin likes pepperoni pizza”.
Q = “Phoebe likes chocolate milkshakes”.
R = “Bonzo likes curly fries”.
The truth table is
P Q R ∼ P ∼ P ∨ Q ∼ Q ∼ Q ∨ R (∼ P ∨ Q) → (∼ Q ∨ R)
T T F F T F F F
The statement is false.
- In each case, determine whether the statement is true or false. Explain your answers.
(a) “If
2 = 1.414, then Silas Hogwinder likes bagels with lox and cream cheese.”
