Construct truth table for ` ~ [ p vv(~ q)] `

Answer

To construct the truth table for the expression `~[p ∨ (~q)]`, we will follow these steps:

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

### Step 2: List the Possible Truth Values
Each variable can either be true (T) or false (F). Therefore, we will have the following combinations for \( p \) and \( q \):

1. \( p = T, q = T \)
2. \( p = T, q = F \)
3. \( p = F, q = T \)
4. \( p = F, q = F \)

### Step 3: Create the Truth Table Structure
We will create a table with columns for \( p \), \( q \), \( \sim q \) (negation of \( q \)), \( p \lor \sim q \) (disjunction of \( p \) and \( \sim q \)), and finally \( \sim [p \lor \sim q] \) (negation of the disjunction).

### Step 4: Fill in the Truth Values for \( q \) and \( \sim q \)
- For \( q = T \), \( \sim q = F \)
- For \( q = F \), \( \sim q = T \)

### Step 5: Calculate \( p \lor \sim q \)
Using the rule of disjunction (logical OR), we determine the values for \( p \lor \sim q \):
- \( T \lor F = T \)
- \( T \lor T = T \)
- \( F \lor F = F \)
- \( F \lor T = T \)

### Step 6: Calculate \( \sim [p \lor \sim q] \)
Now we negate the values obtained in the previous step:
- If \( p \lor \sim q = T \), then \( \sim [p \lor \sim q] = F \)
- If \( p \lor \sim q = F \), then \( \sim [p \lor \sim q] = T \)

### Step 7: Complete the Truth Table
Now we can fill in the truth table with all the values:

| \( p \) | \( q \) | \( \sim q \) | \( p \lor \sim q \) | \( \sim [p \lor \sim q] \) |
|---------|---------|---------------|----------------------|-----------------------------|
| T | T | F | T | F |
| T | F | T | T | F |
| F | T | F | F | T |
| F | F | T | T | F |

### Final Truth Table
The final truth table for the expression `~[p ∨ (~q)]` is:

| \( p \) | \( q \) | \( \sim q \) | \( p \lor \sim q \) | \( \sim [p \lor \sim q] \) |
|---------|---------|---------------|----------------------|-----------------------------|
| T | T | F | T | F |
| T | F | T | T | F |
| F | T | F | F | T |
| F | F | T | T | F |