In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. H \supset(\sim C \vee R) \\ 2. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 3. (H \cdot C) \supset R \text{\_\_\_\_\_\_}$

In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. (G \supset K) \cdot(T \supset K) \\ 2. G \vee T \\ 3. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 4. K \text{\_\_\_\_\_\_}$

In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. \sim G \supset \sim T \\ 2. G \supset N \\ 3. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 4. T \supset N \text{\_\_\_\_\_\_}$

In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. Q \supset(\sim W \supset \sim G) \\ 2. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 3. (Q \bullet G) \supset W \text{\_\_\_\_\_\_}$

In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. C \supset A \\ 2. A \supset C \\ 3. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 4. C \equiv A \text{\_\_\_\_\_\_}$

In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. \sim J \vee M \\ 2. M \supset B \\ 3. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 4. J \supset B \text{\_\_\_\_\_\_}$

In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. E \cdot R \\ 2. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 3. E \equiv R \text{\_\_\_\_\_\_}$

In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. J \supset(M \supset Q) \\ 2. J \bullet M \\ 3. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 4. Q \text{\_\_\_\_\_\_}$

In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. H \supset \sim H \\ 2. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 3. \sim H \text{\_\_\_\_\_\_}$

In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. M \supset(M \supset D) \\ 2. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 3. M \supset D \text{\_\_\_\_\_\_}$

In the following symbolized arguments, derive the line needed to obtain the conclusion (last line), and supply the justification for both lines. $1. \sim S \\ 2. \text{\_\_\_\_\_\_ \_\_\_\_} \\ 3. S \supset K \text{\_\_\_\_\_\_}$

For each of the following lists of premises, derive the indicated conclusion and complete the justification. For tautology, derive a conclusion that is simpler than the premise. $1. (\sim K \vee M) \equiv S \\ 2. T \vee(F \bullet G) \\ 3. R \equiv(N \bullet \sim H) \\ 4. \text{\_\_\_\_\_\_ \_\_\_\_\_, Impl}$

For each of the following lists of premises, derive the indicated conclusion and complete the justification. For tautology, derive a conclusion that is simpler than the premise. $1. (D \bullet H) \vee(\sim D \bullet \sim H) \\ 2. (F \supset J) \bullet (\sim F \supset \sim J) \\ 3. (N \vee T) \bullet(\sim N \vee \sim T) \\ 4. \text{\_\_\_\_\_\_ \_\_\_\_\_, Equiv}$

For each of the following lists of premises, derive the indicated conclusion and complete the justification. For tautology, derive a conclusion that is simpler than the premise. $1. (A \supset \sim C) \cdot(C \supset \sim A) \\ 2. (W \supset \sim T) \cdot(\sim T \supset W) \\ 3. (M \supset \sim E) \cdot(\sim M \supset E) \\ 4. \text{\_\_\_\_\_\_ \_\_\_\_\_, Equiv}$