Use the basic system of derivations along with the attachment rule Adj to establish the following. These repeat entailments from earlier exercises and examples (specifically, b and d of exercise 2.2.x.2, exercises 2 and 4 of 2.3.x, and the example of 2.4.2). They will work best as exercises in the use of Adj if you avoid using Cnj.
| 1. | A ⇒ A ∧ A |
| 2. | A ∧ B, B ∧ C, C ∧ D ⇒ A ∧ D |
| 3. | A ∧ B ⇒ A ∧ (B ∧ A) |
| 4. | A, B ∧ C, D ⇒ (C ∧ (B ∧ A)) ∧ B |
| 5. | A ∧ B ⇒ (B ∧ A) ∧ (A ∧ (B ∧ A)) |