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 in the truth table
Since we have 2 variables, the number of possible combinations of truth values (True or False) is given by \(2^n\), where \(n\) is the number of variables.
Here, \(n = 2\), so \(2^2 = 4\). Therefore, we will have 4 rows in our truth table.

### Step 3: List all combinations of truth values for p and q
We will list all combinations of truth values for p and q:
- 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 4: Calculate the negation of p
Negation of p (denoted as ~p) will be the opposite of the truth value of p:
- Row 1: ~p = False
- Row 2: ~p = False
- Row 3: ~p = True
- Row 4: ~p = True

### Step 5: Write down the values of q
We already have the values of q from our combinations:
- Row 1: q = True
- Row 2: q = False
- Row 3: q = True
- Row 4: q = False

### Step 6: Calculate the conjunction of ~p and q
The conjunction (denoted as ∧) of ~p and q is true only if both ~p and q are true:
- Row 1: ~p = False, q = True → ~p ∧ q = False
- Row 2: ~p = False, q = False → ~p ∧ q = False
- Row 3: ~p = True, q = True → ~p ∧ q = True
- Row 4: ~p = True, q = False → ~p ∧ q = False

### Final Truth Table
Now we can compile all the results into a truth table:

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