Answer the following questions
Using truth table prove that `p vv (q ^^ r) -= (p vv q) ^^ (p vv r)`

Answer

To prove that \( p \lor (q \land r) \equiv (p \lor q) \land (p \lor r) \) using a truth table, we will follow these steps:

### Step 1: Set up the truth table
We need to list all possible truth values for the variables \( p \), \( q \), and \( r \). Since each variable can be either true (T) or false (F), we will have \( 2^3 = 8 \) combinations.

| \( p \) | \( q \) | \( r \) |
|---------|---------|---------|
| 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 2: Calculate \( q \land r \)
Now we will calculate \( q \land r \) for each combination of \( q \) and \( r \).

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

### Step 3: Calculate \( p \lor (q \land r) \)
Next, we will calculate \( p \lor (q \land r) \).

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

### Step 4: Calculate \( p \lor q \) and \( p \lor r \)
Now, we will calculate \( p \lor q \) and \( p \lor r \).

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

### Step 5: Calculate \( (p \lor q) \land (p \lor r) \)
Finally, we will calculate \( (p \lor q) \land (p \lor r) \).

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

### Step 6: Compare the results
Now, we compare the columns for \( p \lor (q \land r) \) and \( (p \lor q) \land (p \lor r) \):

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

Since both columns are identical, we can conclude that:

\[ p \lor (q \land r) \equiv (p \lor q) \land (p \lor r) \]