Exercise K

Answer each of the questions below and justify your answer.

1. Suppose that $\mathcal{A}$ and $\mathcal{B}$ are logically equivalent. What can you say about $\mathcal{A}\leftrightarrow\mathcal{B}$?

2. Suppose that $(\mathcal{A}\wedge\mathcal{B})\rightarrow\mathcal{C}$ is neither a tautology nor a contradiction. What can you say about whether $\mathcal{A},\mathcal{B}\therefore\mathcal{C}$ is valid?

3. Suppose that $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ are jointly unsatisfiable. What can you say about $(\mathcal{A}\wedge\mathcal{B}\wedge\mathcal{C})$?

4. Suppose that $\mathcal{A}$ is a contradiction. What can you say about whether $\mathcal{A},\mathcal{B}\vDash\mathcal{C}$?

5. Suppose that $\mathcal{C}$ is a tautology. What can you say about whether $\mathcal{A},\mathcal{B}\vDash\mathcal{C}$?

6. Suppose that $\mathcal{A}$ and $\mathcal{B}$ are logically equivalent. What can you say about $(\mathcal{A}\vee\mathcal{B})$?

7. Suppose that $\mathcal{A}$ and $\mathcal{B}$ are not logically equivalent. What can you say about $(\mathcal{A}\vee\mathcal{B})$?