If P (r v q) is false then The truth value of (pr) (p(rq))=-------

Let's analyze the given logical statements.

### Given:
- \( P \Rightarrow (r \lor \neg q) \) is false.

### Step 1: Determine the truth values.
The implication \( P \Rightarrow (r \lor \neg q) \) is false if and only if \( P \) is true and \( (r \lor \neg q) \) is false.

- \( P = \text{True} \)
- \( r \lor \neg q = \text{False} \)

For \( r \lor \neg q \) to be false, both \( r \) must be false and \( \neg q \) must be false. If \( \neg q \) is false, then \( q \) is true.

- \( r = \text{False} \)
- \( q = \text{True} \)

### Step 2: Evaluate the truth value of \( (P \Rightarrow r) \Leftrightarrow (P \land (\neg r \Rightarrow q)) \).

1. **Evaluate \( P \Rightarrow r \):**
   - Since \( P = \text{True} \) and \( r = \text{False} \), \( P \Rightarrow r \) is **False**.

2. **Evaluate \( \neg r \Rightarrow q \):**
   - \( \neg r \) is true (since \( r = \text{False} \)).
   - \( \neg r = \text{True} \) and \( q = \text{True} \), so \( \neg r \Rightarrow q \) is **True**.

3. **Evaluate \( P \land (\neg r \Rightarrow q) \):**
   - \( P = \text{True} \) and \( \neg r \Rightarrow q = \text{True} \), so \( P \land (\neg r \Rightarrow q) \) is **True**.

4. **Evaluate \( (P \Rightarrow r) \Leftrightarrow (P \land (\neg r \Rightarrow q)) \):**
   - This is \( \text{False} \Leftrightarrow \text{True} \), which is **False**.

### Final Answer:
The truth value of \( (P \Rightarrow r) \Leftrightarrow (P \land (\neg r \Rightarrow q)) \) is **False**.

Would you like to dive deeper into any part of this reasoning?

### Further Questions:
1. How would the truth value change if \( r \) or \( q \) had different truth values?
2. What are the key differences between implication and biconditional logical operators?
3. How do truth tables help in verifying logical statements like this one?
4. Can you give examples where this kind of logical reasoning might be applied in computer science?
5. How do logical connectives relate to set operations in mathematics?

### Tip:
When analyzing logical statements, always break down each component step-by-step to determine the overall truth value systematically.

Explore the logical statement analysis where P implies (r or not q) is false. Understand the truth values and implications of (P implies r) biconditional to (P and (not r implies q)). Suitable for advanced mathematics enthusiasts.

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