Let p, q and r be three statements. Consider two compound statements
`S_(1):(prArr q)rArr r -= p rArr (p 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 construct truth tables for both statements.

### Step 1: Define the Statements
1. \( S_1: (p \implies (q \implies r)) \equiv (p \implies (p \implies r)) \)
2. \( S_2: (p \iff (q \iff r)) \equiv (p \iff (q \iff r)) \)

### Step 2: Create a Truth Table
We will create a truth table for the variables \( p, q, r \) and evaluate both statements.

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

### Step 3: Evaluate \( S_1 \)
We need to check if \( (p \implies (q \implies r)) \) is equivalent to \( (p \implies (p \implies r)) \).

- From the truth table:
- \( p \implies (q \implies r) \) is T, F, T, T, T, T, T, T
- \( p \implies (p \implies r) \) is T, F, T, F, T, T, T, T

Comparing the two columns:
- They are not equal for all combinations (specifically, they differ when \( p = T, q = F, r = F \)).
- Thus, \( S_1 \) is **False**.

### Step 4: Evaluate \( S_2 \)
Now we check if \( (p \iff (q \iff r)) \) is equivalent to \( (p \iff (q \iff r)) \).

- From the truth table:
- Both expressions yield the same results: T, F, T, F, T, F, T, F.

Since both expressions are equal for all combinations, \( S_2 \) is **True**.

### Final Result
- \( S_1 \) is **False**.
- \( S_2 \) is **True**.

Thus, the answer is:
- \( S_1: F \)
- \( S_2: T \)