A. Use either a derivation or a truth table for each of the following.

1. 1.
   ​

   Show that $A\to [((B\wedge C)\vee D)\to A]$ is a theorem..

2. 2.
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   Show that $A\to (A\to B)$ is not a theorem.

3. 3.
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   Show that the sentence $A\to \neg A$ is not a contradiction.

4. 4.
   ​

   Show that the sentence $A\leftrightarrow \neg A$ is a contradiction.

5. 5.
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   Show that the sentence $\neg (W\to (J\vee J))$ is contingent.

6. 6.
   ​

   Show that the sentence $\neg (X\vee (Y\vee Z))\vee (X\vee (Y\vee Z))$ is not contingent.

7. 7.
   ​

   Show that the sentence $B\to \neg S$ is equivalent to the sentence $\neg \neg B\to \neg S$.

8. 8.
   ​

   Show that the sentence $\neg (X\vee O)$ is not equivalent to the sentence $X\wedge O$

9. 9.
   ​

   Show that the sentences $\neg (A\vee B)$, $C$, $C\to A$ are jointly inconsistent.

10. 10.
    ​

    Show that the sentences $\neg (A\vee B)$, $\neg B$, $B\to A$ are jointly consistent

11. 11.
    ​

    Show that $\neg (A\vee (B\vee C))$ $\therefore$$\neg C$ is valid.

12. 12.
    ​

    Show that $\neg (A\wedge (B\vee C))$ $\therefore$$\neg C$ is invalid.

B. Use either a derivation or a truth table for each of the following.

1. 1.
   ​

   Show that $A\to (B\to A)$ is a theorem.

2. 2.
   ​

   Show that $\neg (((N\leftrightarrow Q)\vee Q)\vee N)$ is not a theorem.

3. 3.
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   Show that $Z\vee (\neg Z\leftrightarrow Z)$ is contingent.

4. 4.
   ​

   show that $(L\leftrightarrow ((N\to N)\to L))\vee H$ is not contingent.

5. 5.
   ​

   Show that $(A\leftrightarrow A)\wedge (B\wedge \neg B)$ is a contradiction.

6. 6.
   ​

   Show that $(B\leftrightarrow (C\vee B))$ is not a contradiction.

7. 7.
   ​

   Show that $((\neg X\leftrightarrow X)\vee X)$ is equivalent to $X$.

8. 8.
   ​

   Show that $F\wedge (K\wedge R)$ is not equivalent to $(F\leftrightarrow (K\leftrightarrow R))$.

9. 9.
   ​

   Show that the sentences $\neg (W\to W)$, $(W\leftrightarrow W)\wedge W$, $E\vee (W\to \neg (E\wedge W))$ are jointly inconsistent.

10. 10.
    ​

    Show that the sentences $\neg R\vee C$, $(C\wedge R)\to \neg R$, $(\neg (R\vee C)\to R)$ are jointly consistent.

11. 11.
    ​

    Show that $\neg \neg (C\leftrightarrow \neg C),((G\vee C)\vee G)\therefore ((G\to C)\wedge G)$ is valid.

12. 12.
    ​

    Show that $\neg \neg L,(C\to \neg L)\to C)\therefore \neg C$ is invalid.