When does the value of the statement ` (p ^^ r) harr ( r ^^ q) ` become false ?

Options

A. p is T , q is F
B. p is T, q is T and r is F
C. p is F, q is F and r is F
D. none of these

Answer

To determine when the statement \( (p \land r) \leftrightarrow (r \land q) \) becomes false, we will construct a truth table and analyze the conditions under which the statement evaluates to false.

### Step-by-Step Solution:

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

2. **List Possible Combinations**: Each variable can either be true (T) or false (F). Therefore, we will list all possible combinations of truth values for \( p \), \( q \), and \( r \):
- \( (F, F, F) \)
- \( (F, F, T) \)
- \( (F, T, F) \)
- \( (F, T, T) \)
- \( (T, F, F) \)
- \( (T, F, T) \)
- \( (T, T, F) \)
- \( (T, T, T) \)

3. **Calculate \( p \land r \)**: This column represents the conjunction (AND) of \( p \) and \( r \):
- \( F \land F = F \)
- \( F \land T = F \)
- \( F \land F = F \)
- \( F \land T = F \)
- \( T \land F = F \)
- \( T \land T = T \)
- \( T \land F = F \)
- \( T \land T = T \)

4. **Calculate \( r \land q \)**: This column represents the conjunction of \( r \) and \( q \):
- \( F \land F = F \)
- \( F \land T = F \)
- \( T \land F = F \)
- \( T \land T = T \)
- \( F \land F = F \)
- \( T \land T = T \)
- \( F \land T = F \)
- \( T \land T = T \)

5. **Calculate \( (p \land r) \leftrightarrow (r \land q) \)**: This column represents the biconditional (if and only if) between the two previous columns. The biconditional is true if both sides are the same (both true or both false):
- \( F \leftrightarrow F = T \)
- \( F \leftrightarrow F = T \)
- \( F \leftrightarrow F = T \)
- \( F \leftrightarrow T = F \)
- \( F \leftrightarrow F = T \)
- \( T \leftrightarrow T = T \)
- \( F \leftrightarrow F = T \)
- \( T \leftrightarrow T = T \)

6. **Identify False Outcomes**: From our calculations, we see that the biconditional \( (p \land r) \leftrightarrow (r \land q) \) is false in the following scenario:
- When \( p = F \), \( q = T \), and \( r = T \).

### Conclusion:
The value of the statement \( (p \land r) \leftrightarrow (r \land q) \) becomes false when \( p \) is false, \( q \) is true, and \( r \) is true.