If p, q, r be anY three. statements
STATEMENT-1 : `pvv(q^^r)hArr(pvvq)^^(pvvr)`
and
STATEMENT-2 : `pvv(q^^r)=(pvvq)^^(pvvr)`

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we need to analyze the two statements given and check their validity using a truth table.

### Step 1: Define the Statements
- **Statement 1**: \( p \lor (q \land r) \iff (p \lor q) \land (p \lor r) \)
- **Statement 2**: \( p \lor (q \land r) = (p \lor q) \land (p \lor r) \)

### Step 2: Create a Truth Table
We will create a truth table for the variables \( p, q, r \) which can take values True (T) or False (F). Since there are three variables, we will have \( 2^3 = 8 \) possible combinations of truth values.

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

### Step 3: Evaluate the Columns
- For \( q \land r \): True only when both \( q \) and \( r \) are True.
- For \( p \lor (q \land r) \): True if either \( p \) is True or \( q \land r \) is True.
- For \( p \lor q \): True if either \( p \) or \( q \) is True.
- For \( p \lor r \): True if either \( p \) or \( r \) is True.
- For \( (p \lor q) \land (p \lor r) \): True only when both \( p \lor q \) and \( p \lor r \) are True.

### Step 4: Compare the Results
Now we will compare the columns for \( p \lor (q \land r) \) and \( (p \lor q) \land (p \lor r) \).

From the truth table:
- The column for \( p \lor (q \land r) \) is: T, T, T, T, T, F, F, F
- The column for \( (p \lor q) \land (p \lor r) \) is: T, T, T, T, T, F, F, F

### Step 5: Conclusion
Since both columns are identical for all combinations of truth values, we can conclude that:
- **Statement 1 is true**: \( p \lor (q \land r) \iff (p \lor q) \land (p \lor r) \)
- **Statement 2 is also true**: \( p \lor (q \land r) = (p \lor q) \land (p \lor r) \)

Thus, both statements are valid.