construct truth table for `~(~p ^^ ~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: Determine the Number of Rows
Since we have 2 variables, the number of possible combinations of truth values is \( 2^2 = 4 \). Therefore, we will have 4 rows in our truth table.

### Step 3: Create the Truth Table Structure
We will create columns for \( p \), \( q \), \( \sim p \) (negation of \( p \)), \( \sim q \) (negation of \( q \)), \( \sim p \land \sim q \) (conjunction of negation of \( p \) and negation of \( q \)), and finally \( \sim(\sim p \land \sim q) \) (negation of the conjunction).

### Step 4: Fill in the Values for \( p \) and \( q \)
We will fill in the truth values for \( p \) and \( q \):

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

### Step 5: Calculate \( \sim p \) and \( \sim q \)
Next, we will calculate the negations of \( p \) and \( q \):

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

### Step 6: Calculate \( \sim p \land \sim q \)
Now we will calculate the conjunction of \( \sim p \) and \( \sim q \):

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

### Step 7: Calculate \( \sim(\sim p \land \sim q) \)
Finally, we will calculate the negation of the conjunction:

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

### Final Truth Table
The complete truth table for the expression `~(~p ∧ ~q)` is:

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