construct truth table for `(~p ^^ ~q) v(p ^^ ~q)`

Answer

To construct the truth table for the expression `(~p ∧ ~q) ∨ (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 \( 2^n = 2^2 = 4 \). Therefore, we will have 4 rows in our truth table.

### Step 3: Create the Truth Table Structure
We will create a table with the following columns:
1. \( p \)
2. \( q \)
3. \( \neg p \) (Negation of \( p \))
4. \( \neg q \) (Negation of \( q \))
5. \( \neg p \land \neg q \) (Conjunction of \( \neg p \) and \( \neg q \))
6. \( p \land \neg q \) (Conjunction of \( p \) and \( \neg q \))
7. \( (\neg p \land \neg q) \lor (p \land \neg q) \) (Disjunction of the previous two results)

### Step 4: Fill in the Values
Now we will fill in the truth values for each variable and expression.

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

### Step 5: Analyze the Results
- In the first row, both \( p \) and \( q \) are true, so both \( \neg p \) and \( \neg q \) are false. Therefore, \( \neg p \land \neg q \) is false, and \( p \land \neg q \) is also false, resulting in a final value of false.
- In the second row, \( p \) is true and \( q \) is false, making \( \neg p \) false and \( \neg q \) true. Hence, \( \neg p \land \neg q \) is false, but \( p \land \neg q \) is true, resulting in a final value of true.
- In the third row, \( p \) is false and \( q \) is true, so \( \neg p \) is true and \( \neg q \) is false. Both conjunctions yield false, resulting in a final value of false.
- In the fourth row, both \( p \) and \( q \) are false, making both \( \neg p \) and \( \neg q \) true. Thus, \( \neg p \land \neg q \) is true, while \( p \land \neg q \) is false, resulting in a final value of true.

### Final Truth Table
The final truth table is:

| \( p \) | \( q \) | \( \neg p \) | \( \neg q \) | \( \neg p \land \neg q \) | \( p \land \neg q \) | Final Result |
|---------|---------|--------------|--------------|-----------------------------|-----------------------|--------------|
| T | T | F | F | F | F | F |
| T | F | F | T | F | T | T |
| F | T | T | F | F | F | F |
| F | F | T | T | T | F | T |