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
- q

### Step 2: Determine the Number of Rows
Since we have 2 variables, the number of possible combinations of truth values (True or False) is \(2^n\), where \(n\) is the number of variables. Thus, for 2 variables, we have:
\[ 2^2 = 4 \]
This means 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. p ∧ q (conjunction of p and q)
4. ~(p ∧ 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. The combinations will be:
- Row 1: p = True, q = True
- Row 2: p = True, q = False
- Row 3: p = False, q = True
- Row 4: p = False, q = False

### Step 5: Calculate p ∧ q
Now, we will calculate the conjunction (AND operation) for each row:
- Row 1: True ∧ True = True
- Row 2: True ∧ False = False
- Row 3: False ∧ True = False
- Row 4: False ∧ False = False

### Step 6: Calculate ~(p ∧ q)
Next, we will calculate the negation of the conjunction for each row:
- Row 1: ~(True) = False
- Row 2: ~(False) = True
- Row 3: ~(False) = True
- Row 4: ~(False) = True

### Step 7: Complete the Truth Table
Now we can complete the truth table:

| p | q | p ∧ q | ~(p ∧ q) |
|-------|-------|-------|----------|
| True | True | True | False |
| True | False | False | True |
| False | True | False | True |
| False | False | False | True |

### Conclusion
The truth table for the expression `~(p ∧ q)` is complete.

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