Puzzle
The Dodo says that the Hatter tells lies.
The Hatter says that the March Hare tells lies.
The March Hare says that both the Dodo and the Hatter tell lies.
Who is telling the truth?
Hint: Consider the three different cases in which only one of the characters is telling the truth. In only one of these cases can all three of the statements be true.
1. Write truth tables for the following statements
(a) ∼ (p ∧ q) ∨ (p ∧ q)
(b) p ∧ (∼ q ∨ r)
(c) (p ∨ q) ∨ (∼ p ∧ q) → q
(d) [(p ∧ ∼ q) ∨ (∼ r ∧ p)] → (r ∨ ∼ q)
(e) ∼ (p ∨ r) ↔ q
2. Determine the truth value of the statement given that p is a true (T) statement, q is a false (F) statement, and r is a true (T) statement.
(a) (p ∧ q) ∨ ∼ r
(b) [(p ∧ ∼ q) ∨ ∼ r] ∧ (p ∧ r)
(c) (p ∧ ∼ q) → r
(d) (p ∨ r) → (q ∧ ∼ r)
(e) (p ∧ ∼ p) ↔ (p → q)
3. Determine whether the statements are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer.
(a) (p ∨ q) ∧ r and (p ∨ r) ∧ (q ∧ r)
(b) p →∼ r and r ∨ ∼ p
(c) p → q and q → p
(d) p → (q ∨ r) and (p → q) ∨ (p → r)
4. Use truth tables to determine which of the statement forms are tautologies and which are contradiction.
(a) (p ∧ ∼ q) ∧ (∼ p ∨ q)
(b) (∼ p ∨ q) ∨ ( p ∧ ∼ q)
(c) (p ∧ q) ∨ (∼ p ∨ ∼ q)
5. Write each argument in symbolic form, using the letters p, q or r.
(a) If the demand for face masks increase, the manufacturer produces more face masks. The demand for face masks does not increase. Therefore, the manufacturer does not produce more
face masks.
(b) If it rains, the soil is wet. It does not rain. Therefore the soil is not wet.
6. Use a truth table to determine whether the argument is valid or invalid.
p → q
q
∴ p
The Dodo says that the Hatter tells lies.
The Hatter says that the March Hare tells lies.
The March Hare says that both the Dodo and the Hatter tell lies.
Who is telling the truth?
Hint: Consider the three different cases in which only one of the characters is telling the truth. In only one of these cases can all three of the statements be true.
1. Write truth tables for the following statements
(a) ∼ (p ∧ q) ∨ (p ∧ q)
(b) p ∧ (∼ q ∨ r)
(c) (p ∨ q) ∨ (∼ p ∧ q) → q
(d) [(p ∧ ∼ q) ∨ (∼ r ∧ p)] → (r ∨ ∼ q)
(e) ∼ (p ∨ r) ↔ q
2. Determine the truth value of the statement given that p is a true (T) statement, q is a false (F) statement, and r is a true (T) statement.
(a) (p ∧ q) ∨ ∼ r
(b) [(p ∧ ∼ q) ∨ ∼ r] ∧ (p ∧ r)
(c) (p ∧ ∼ q) → r
(d) (p ∨ r) → (q ∧ ∼ r)
(e) (p ∧ ∼ p) ↔ (p → q)
3. Determine whether the statements are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer.
(a) (p ∨ q) ∧ r and (p ∨ r) ∧ (q ∧ r)
(b) p →∼ r and r ∨ ∼ p
(c) p → q and q → p
(d) p → (q ∨ r) and (p → q) ∨ (p → r)
4. Use truth tables to determine which of the statement forms are tautologies and which are contradiction.
(a) (p ∧ ∼ q) ∧ (∼ p ∨ q)
(b) (∼ p ∨ q) ∨ ( p ∧ ∼ q)
(c) (p ∧ q) ∨ (∼ p ∨ ∼ q)
5. Write each argument in symbolic form, using the letters p, q or r.
(a) If the demand for face masks increase, the manufacturer produces more face masks. The demand for face masks does not increase. Therefore, the manufacturer does not produce more
face masks.
(b) If it rains, the soil is wet. It does not rain. Therefore the soil is not wet.
6. Use a truth table to determine whether the argument is valid or invalid.
p → q
q
∴ p
Answers & Comments
Answer:
1.
(a) ∼ (p ∧ q) ∨ (p ∧ q) - This statement is a tautology, meaning it is always true regardless of the truth values of p and q.
(b) p ∧ (∼ q ∨ r) - The truth value of this statement depends on the truth values of p, q, and r.
(c) (p ∨ q) ∨ (∼ p ∧ q) → q - This statement is also a tautology, always true regardless of the truth values of p and q.
(d) [(p ∧ ∼ q) ∨ (∼ r ∧ p)] → (r ∨ ∼ q) - The truth value of this statement depends on the truth values of p, q, and r.
(e) ∼ (p ∨ r) ↔ q - The truth value of this statement depends on the truth values of p and r.
2. Given that p is true (T), q is false (F), and r is true (T):
(a) (p ∧ q) ∨ ∼ r - The truth value of this statement is false (F).
(b) [(p ∧ ∼ q) ∨ ∼ r] ∧ (p ∧ r) - The truth value of this statement is true (T).
(c) (p ∧ ∼ q) → r - The truth value of this statement is true (T).
(d) (p ∨ r) → (q ∧ ∼ r) - The truth value of this statement is true (T).
(e) (p ∧ ∼ p) ↔ (p → q) - The truth value of this statement is false (F).
3.
(a) (p ∨ q) ∧ r and (p ∨ r) ∧ (q ∧ r) - These statements are not logically equivalent. The truth tables for each statement will show different evaluation results.
(b) p →∼ r and r ∨ ∼ p - These statements are not logically equivalent. The truth tables for each statement will show different evaluation results.
(c) p → q and q → p - These statements are logically equivalent. The truth tables for each statement will show matching evaluation results.
(d) p → (q ∨ r) and (p → q) ∨ (p → r) - These statements are logically equivalent. The truth tables for each statement will show matching evaluation results.
4.
(a) (p ∧ ∼ q) ∧ (∼ p ∨ q) - This statement form is a contradiction, always false regardless of the truth values of p and q.
(b) (∼ p ∨ q) ∨ (p ∧ ∼ q) - This statement form is a tautology, always true regardless of the truth values of p and q.
(c) (p ∧ q) ∨ (∼ p ∨ ∼ q) - This statement form is also a tautology, always true regardless of the truth values of p and q.
5.
(a) If the demand for face masks increase (p), the manufacturer produces more face masks (q).
Symbolic form: p → q
(b) If it rains (p), the soil is wet (q).
Symbolic form: p → q
6. Using a truth table to determine the validity of the argument:
p | q | (p → q) | q | ∴ p
--------------------------------
True | True | True | T | T
True | False | False | F | T
False | True | True | T | F
False | False | True | F | F
Based on the truth table, the argument is invalid. The conclusion does not follow from the premises.
Correct me if I am wrong.