4.2.x. Exercises
| 1. |
Use derivations to establish each of the claims of entailment and equivalence shown below. (Remember that claims of equivalence require derivations in both directions.) |
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| a. | A ∧ B ⇒ A ∨ B | |
| b. | A ∧ B ⇒ B ∨ C | |
| c. | A ∨ B, ¬ A ⇒ B | |
| d. | A ∨ (A ∧ B) ⇒ A | |
| e. | A ∨ B, ¬ (A ∧ C), ¬ (B ∧ C) ⇒ ¬ C | |
| f. | A ∧ (B ∨ C) ⇒ (A ∧ B) ∨ C | |
| g. | A ∨ B, C ⇒ (A ∧ C) ∨ (B ∧ C) | |
| h. | A ∨ B, ¬ A ∨ C ⇒ B ∨ C | |
| i. | A ⇔ (A ∧ B) ∨ (A ∧ ¬ B) | |
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2. |
Use derivations to establish each of the claims of equivalence below. | |
| a. | A ∨ A ⇔ A | |
| b. | A ∨ B ⇔ B ∨ A | |
| c. | A ∨ (B ∨ C) ⇔ (A ∨ B) ∨ C | |
| d. | A ∨ (B ∧ ¬ B) ⇔ A | |
| e. | ¬ (A ∨ B) ⇔ ¬ A ∧ ¬ B | |
| f. | ¬ (A ∧ B) ⇔ ¬ A ∨ ¬ B | |
| 3. |
Use derivations to check each of the claims below; if a derivation indicates that a claim fails, present a counterexample that divides an open gap. |
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| a. | A ∨ B, A ⇒ ¬ B | |
| b. | A ∨ (B ∧ C) ⇔ (A ∨ B) ∧ C | |
| c. | ¬ (A ∨ B) ⇔ ¬ A ∨ ¬ B | |