Construct the truth table for the compound proposition `q rArr [~pvvq]`

Answer

To construct the truth table for the compound proposition \( q \rightarrow [\neg p \vee q] \), we will follow these steps:

### Step 1: Identify the variables
The variables in the proposition are \( p \) and \( q \). We will consider all possible truth values for these variables.

### Step 2: Create the columns for the variables
We will create columns for \( p \) and \( q \) and list all possible combinations of truth values (True or False).

| \( p \) | \( q \) |
|---------|---------|
| T | T |
| T | F |
| F | T |
| F | F |

### Step 3: Calculate \( \neg p \)
Next, we will calculate the negation of \( p \) (\( \neg p \)).

| \( p \) | \( q \) | \( \neg p \) |
|---------|---------|---------------|
| T | T | F |
| T | F | F |
| F | T | T |
| F | F | T |

### Step 4: Calculate \( \neg p \vee q \)
Now, we will calculate the disjunction \( \neg p \vee q \).

| \( p \) | \( q \) | \( \neg p \) | \( \neg p \vee q \) |
|---------|---------|---------------|----------------------|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |

### Step 5: Calculate \( q \rightarrow [\neg p \vee q] \)
Finally, we will calculate the implication \( q \rightarrow [\neg p \vee q] \).

| \( p \) | \( q \) | \( \neg p \) | \( \neg p \vee q \) | \( q \rightarrow [\neg p \vee q] \) |
|---------|---------|---------------|----------------------|-------------------------------------|
| T | T | F | T | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | F | T | T | T |

### Final Truth Table
The final truth table for the compound proposition \( q \rightarrow [\neg p \vee q] \) is:

| \( p \) | \( q \) | \( \neg p \) | \( \neg p \vee q \) | \( q \rightarrow [\neg p \vee q] \) |
|---------|---------|---------------|----------------------|-------------------------------------|
| T | T | F | T | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | F | T | T | T |