construct truth table for ` p vv(~q)`

Answer

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

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

### Step 2: List all possible truth values
Each variable can be either True (T) or False (F). Therefore, we will list all combinations of truth values for \( p \) and \( q \):
- \( p = T, q = T \)
- \( p = T, q = F \)
- \( p = F, q = T \)
- \( p = F, q = F \)

### Step 3: Create a column for \( \neg q \)
The negation of \( q \) (denoted as \( \neg q \)) will be the opposite truth value of \( q \):
- If \( q = T \), then \( \neg q = F \)
- If \( q = F \), then \( \neg q = T \)

### Step 4: Create a column for \( p \lor \neg q \)
Now we will evaluate \( p \lor \neg q \) (which means \( p \) OR \( \neg q \)):
- The OR operation results in True if at least one of the operands is True.

### Step 5: Fill in the truth table
Now we can fill in the truth table based on the values we identified:

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

### Final Truth Table
The completed truth table for \( p \lor \neg q \) is:

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