1. Analyze the logical forms of the following statements. Use A to represent "Alice has a dog," B to represent "Bob has a dog," and C to represent "Carol has a cat" to write each as a symbolic statement.
a. Either Alice or Bob has a dog.
b. Neither Alice nor Bob hasa dog, but Carol has a cat.
c. Either Alice has a dog and Carol has a cat, or Bob has a dog and Carol does not have a cat.
2. If D stands for "Doug is tall" and E stands for "Edie is short," what English sentences are represented by the following expressions?
a. (D ∧ E) ∨ ¬D
b. (D ∨¬E) ∧ ¬(D ∧ E)
c. ¬D ∧ [ (E ∧ D) ∨ ¬E ]
3. Make a truth table for the following formula.
(G ∨ ¬H) ∧¬(G ∧ L)
4. Use truth tables to determine which of the following formulas are equivalent to each other.
a. (J ∧ K) ∨ (¬J ∧¬K)
b. J ∨ K
c. J ∧¬K
d. ¬(¬J ∨ K)
e. (J ∧¬K) ∨ K
5. Use truth tables to determine which of the following formulas are tautologies, which are contradictions, and which are neither.
a. (M ∧ ¬N) ∨ (¬M ∧ N)
b. (M ∧ ¬N) ∧ (¬M ∧ N)
c. (¬M ∧ ¬N) ∨ (¬M ∨ N) ∨ (M ∧ ¬N)
6. Use the laws stated in the text to find simpler formulas equivalent to these formulas. Explain the reasoning that you used to find your solution.
a. ¬[ ¬Q ∨ (¬P ∧ Q) ]
b. ((P ∧ Q) ∧ ¬R) ∨ [P ∧ ¬(Q ∨ R)]