| 1. | The following arguments are not formally valid. In each case, use a derivation to show this and present a counterexample that the derivation leads you to. | |
| a. | ¬ B / ¬ (A ∧ ¬ B) | |
| b. | ¬ (A ∧ B) / ¬ A ∧ ¬ B | |
| c. | ¬ (A ∧ B), ¬ (B ∧ C) / ¬ (A ∧ C) | |
| 2. | Use derivations to check the following claims of entailment. If the claim fails, present a counterexample that the derivation leads you to. | |
| a. | ¬ (A ∧ ¬ B) ⇒ B | |
| b. | ¬ (A ∧ B) ⇒ ¬ (B ∧ A) | |
| c. | ¬ (A ∧ ¬ B) ⇒ ¬ (B ∧ ¬ A) | |
| d. | ¬ (A ∧ B), ¬ (B ∧ C), B ⇒ ¬ A ∧ ¬ C | |
| e. | ¬ (A ∧ ¬ (B ∧ ¬ (C ∧ ¬ D))) ⇒ ¬ (A ∧ ¬ (B ∧ D)) | |