Let p, q and r be three statements. Consider two compound statements `S_(1):(prArr q)rArr r -= p rArr (q rArr r)` `S_(2):(p hArr q)hArrr -=phArr (q hArr r)` State in order, whether `S_(1), S_(2)` are true of false. (where, T represents true F represents false)

Options

A. TT
B. TF
C. FT
D. FF

Answer

To determine the truth values of the compound statements \( S_1 \) and \( S_2 \), we will analyze each statement step by step.

### Step 1: Analyze \( S_1 \)

The statement \( S_1 \) is given as:
\[ S_1: (p \land q) \to r \quad \text{is equivalent to} \quad p \to (q \to r) \]

We will use a truth table to evaluate this statement.

#### Truth Table for \( S_1 \)

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

Now we compare the columns for \( (p \land q) \to r \) and \( p \to (q \to r) \):
- The truth values for \( (p \land q) \to r \) are: T, F, T, T, T, T, T, T
- The truth values for \( p \to (q \to r) \) are: T, F, T, F, T, T, T, T

Since these two columns are not identical, \( S_1 \) is **False**.

### Step 2: Analyze \( S_2 \)

The statement \( S_2 \) is given as:
\[ S_2: (p \leftrightarrow q) \to r \quad \text{is equivalent to} \quad p \leftrightarrow (q \leftrightarrow r) \]

Again, we will create a truth table for this statement.

#### Truth Table for \( S_2 \)

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

Now we compare the columns for \( (p \leftrightarrow q) \to r \) and \( p \leftrightarrow (q \leftrightarrow r) \):
- The truth values for \( (p \leftrightarrow q) \to r \) are: T, F, T, T, T, T, T, F
- The truth values for \( p \leftrightarrow (q \leftrightarrow r) \) are: T, F, F, T, F, T, F, T

Since these two columns are not identical, \( S_2 \) is also **False**.

### Final Conclusion

- \( S_1 \) is **False**
- \( S_2 \) is **False**