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 there are 2 variables, the number of possible combinations of truth values is \( 2^n \) where \( n \) is the number of variables. Here, \( n = 2 \), so \( 2^2 = 4 \). Thus, we will have 4 rows in our truth table.

### Step 3: Create the Table Structure
We will create a table with the following columns:
1. \( p \)
2. \( q \)
3. \( \sim p \) (negation of \( p \))
4. \( \sim q \) (negation of \( q \))
5. \( \sim p \land \sim q \) (conjunction of negation of \( p \) and negation of \( q \))

### Step 4: Fill in the Values for \( p \) and \( q \)
We will assign the truth values for \( p \) and \( q \) as follows:
- Row 1: \( p = \text{True}, q = \text{True} \)
- Row 2: \( p = \text{True}, q = \text{False} \)
- Row 3: \( p = \text{False}, q = \text{True} \)
- Row 4: \( p = \text{False}, q = \text{False} \)

### Step 5: Calculate \( \sim p \) and \( \sim q \)
Now we will calculate the negations:
- For \( \sim p \):
- Row 1: \( \sim p = \text{False} \)
- Row 2: \( \sim p = \text{False} \)
- Row 3: \( \sim p = \text{True} \)
- Row 4: \( \sim p = \text{True} \)

- For \( \sim q \):
- Row 1: \( \sim q = \text{False} \)
- Row 2: \( \sim q = \text{True} \)
- Row 3: \( \sim q = \text{False} \)
- Row 4: \( \sim q = \text{True} \)

### Step 6: Calculate \( \sim p \land \sim q \)
Now we will calculate the conjunction of \( \sim p \) and \( \sim q \):
- Row 1: \( \sim p \land \sim q = \text{False} \land \text{False} = \text{False} \)
- Row 2: \( \sim p \land \sim q = \text{False} \land \text{True} = \text{False} \)
- Row 3: \( \sim p \land \sim q = \text{True} \land \text{False} = \text{False} \)
- Row 4: \( \sim p \land \sim q = \text{True} \land \text{True} = \text{True} \)

### Final Truth Table
Now we can summarize our findings in the truth table:

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