`p ^^ ( q ^^ r)` is logically equivalent to

Options

A. `p vv ( q ^^r)`
B. ` ( p ^^ q) ^^ r`
C. ` ( p vv q) vv r`
D. ` p to ( q ^^ r)`

Answer

To determine the logical equivalence of the expression \( p \land (q \land r) \), we can follow these steps:

### Step-by-Step Solution:

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

2. **Determine the Number of Possibilities**:
Since there are three variables, the total number of combinations of truth values (True or False) is \( 2^3 = 8 \).

3. **Construct the Truth Table**:
We will create a truth table for the expression \( p \land (q \land r) \).

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

4. **Analyze the Truth Table**:
From the truth table, we can see that \( p \land (q \land r) \) is only true when all three variables \( p \), \( q \), and \( r \) are true. In all other cases, the expression evaluates to false.

5. **Check Logical Equivalence**:
We need to check which of the following expressions is logically equivalent to \( p \land (q \land r) \):
- \( p \lor (q \land r) \)
- \( p \land q \land r \)
- \( p \lor q \lor r \)
- \( p \implies (q \land r) \)

We will construct truth tables for each of these expressions and compare them to the truth table of \( p \land (q \land r) \).

6. **Truth Table for \( p \land q \land r \)**:
| \( p \) | \( q \) | \( r \) | \( p \land q \land r \) |
|---------|---------|---------|--------------------------|
| T | T | T | T |
| T | T | F | F |
| T | F | T | F |
| T | F | F | F |
| F | T | T | F |
| F | T | F | F |
| F | F | T | F |
| F | F | F | F |

The truth table for \( p \land q \land r \) matches exactly with that of \( p \land (q \land r) \).

7. **Conclusion**:
Therefore, \( p \land (q \land r) \) is logically equivalent to \( p \land q \land r \).

### Final Answer:
\( p \land (q \land r) \) is logically equivalent to \( p \land q \land r \).