Construct the truth table for the compound proposition `(~pvvq)`

Answer

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

### Step 1: Identify the Variables
We have two variables in our proposition: \( p \) and \( q \).

### Step 2: List All Possible Truth Values
Since there are two variables, we will have \( 2^2 = 4 \) combinations of truth values. The combinations for \( p \) and \( q \) are as follows:

- \( p = \text{True}, q = \text{True} \)
- \( p = \text{True}, q = \text{False} \)
- \( p = \text{False}, q = \text{True} \)
- \( p = \text{False}, q = \text{False} \)

### Step 3: Create the Table Structure
We will create a table with the following columns:
1. \( p \)
2. \( q \)
3. \( \neg p \) (negation of \( p \))
4. \( \neg p \lor q \) (the compound proposition)

### Step 4: Fill in the Truth Values for \( p \) and \( q \)
Now we fill in the truth values for \( p \) and \( q \):

| \( p \) | \( q \) |
|----------|----------|
| True | True |
| True | False |
| False | True |
| False | False |

### Step 5: Calculate \( \neg p \)
Next, we calculate the negation of \( p \):

- If \( p \) is True, \( \neg p \) is False.
- If \( p \) is False, \( \neg p \) is True.

So, we fill in the values for \( \neg p \):

| \( p \) | \( q \) | \( \neg p \) |
|----------|----------|---------------|
| True | True | False |
| True | False | False |
| False | True | True |
| False | False | True |

### Step 6: Calculate \( \neg p \lor q \)
Now we compute the compound proposition \( \neg p \lor q \):

- The logical OR (\( \lor \)) is True if at least one of the operands is True.

| \( p \) | \( q \) | \( \neg p \) | \( \neg p \lor q \) |
|----------|----------|---------------|-----------------------|
| True | True | False | True |
| True | False | False | False |
| False | True | True | True |
| False | False | True | True |

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

| \( p \) | \( q \) | \( \neg p \) | \( \neg p \lor q \) |
|----------|----------|---------------|-----------------------|
| True | True | False | True |
| True | False | False | False |
| False | True | True | True |
| False | False | True | True |