construct truth table for `p ^^ (q vv r)`

Answer

To construct the truth table for the logical expression \( p \land (q \lor r) \), we will follow these steps:

### Step 1: Identify the Variables
We have three variables: \( p \), \( q \), and \( r \).

### Step 2: Determine the Number of Rows
Since we have 3 variables, the number of possible combinations of truth values is \( 2^n \), where \( n \) is the number of variables. Thus, we have \( 2^3 = 8 \) rows in our truth table.

### Step 3: Set Up the Truth Table Structure
We will create a table with the following columns:
1. \( p \)
2. \( q \)
3. \( r \)
4. \( q \lor r \) (disjunction of \( q \) and \( r \))
5. \( p \land (q \lor r) \) (conjunction of \( p \) and \( (q \lor r) \))

### Step 4: Fill in the Values for \( p \), \( q \), and \( r \)
We will fill in the truth values for \( p \), \( q \), and \( r \) in the table. The values will alternate between True (T) and False (F) as follows:

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

### Step 5: Calculate \( q \lor r \)
Now we will compute the values for \( q \lor r \). The disjunction is true if at least one of \( q \) or \( r \) is true.

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

### Step 6: Calculate \( p \land (q \lor r) \)
Finally, we will compute the values for \( p \land (q \lor r) \). The conjunction is true only if both \( p \) and \( (q \lor r) \) are true.

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

### Final Truth Table
Combining all the columns, we have the final truth table:

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