-----

5) If (pvQ) (RR) is True , then  the truth value of a)RQ is-----

                                                                                     b)PR is-----  

                                                                                         c)(QR)R is---------

                                                                                        d)(QR)P is-----


Given that \((\neg p \lor Q) \Rightarrow (R \land \neg R)\) is **true**, we need to determine the truth values of the statements \(R \Rightarrow Q\), \(P \land \neg R\), \((Q \Rightarrow R) \land R\), and \((Q \land R) \Rightarrow P\).

### Step 1: Analyze the statement \((\neg p \lor Q) \Rightarrow (R \land \neg R)\)

First, notice that the expression \(R \land \neg R\) represents a contradiction because \(R\) and \(\neg R\) cannot both be true simultaneously. A contradiction is always false. Thus, \(R \land \neg R\) is false.

For the conditional statement \((\neg p \lor Q) \Rightarrow (R \land \neg R)\) to be true, the antecedent \(\neg p \lor Q\) must be false. This is because a conditional statement with a false consequent (which we have here) is true only when the antecedent is false.

### Step 2: Analyze \(\neg p \lor Q\)
The disjunction \(\neg p \lor Q\) is false only when both \(\neg p\) and \(Q\) are false. 
- \(\neg p\) is false implies \(P\) is true.
- \(Q\) is false.

So, we have:
- \(P\) is true.
- \(Q\) is false.
- \(R\) can be either true or false (since \(R \land \neg R\) forces no specific value on \(R\)).

### Step 3: Determine the truth values for each statement

a) \(R \Rightarrow Q\):

\(R \Rightarrow Q\) is true if \(R\) is false or \(Q\) is true. Since \(Q\) is false, the truth value of \(R \Rightarrow Q\) depends on \(R\):
- If \(R\) is true, \(R \Rightarrow Q\) is false.
- If \(R\) is false, \(R \Rightarrow Q\) is true.

b) \(P \land \neg R\):

\(P \land \neg R\) is true if both \(P\) is true and \(\neg R\) is true (i.e., \(R\) is false). Since \(P\) is true:
- If \(R\) is true, \(P \land \neg R\) is false.
- If \(R\) is false, \(P \land \neg R\) is true.

c) \((Q \Rightarrow R) \land R\):

\(Q \Rightarrow R\) is true if \(Q\) is false (which it is) or \(R\) is true. Since \(Q\) is false, \(Q \Rightarrow R\) is true regardless of \(R\). 
- If \(R\) is true, \((Q \Rightarrow R) \land R\) is true.
- If \(R\) is false, \((Q \Rightarrow R) \land R\) is false.

d) \((Q \land R) \Rightarrow P\):

\(Q \land R\) is true only if both \(Q\) and \(R\) are true. Since \(Q\) is false, \(Q \land R\) is false, and the conditional \((Q \land R) \Rightarrow P\) is true regardless of \(P\).

### Summary of Truth Values:
The truth values depend on whether \(R\) is true or false:
- If \(R\) is **true**:
  - a) \(R \Rightarrow Q\) is **false**.
  - b) \(P \land \neg R\) is **false**.
  - c) \((Q \Rightarrow R) \land R\) is **true**.
  - d) \((Q \land R) \Rightarrow P\) is **true**.

- If \(R\) is **false**:
  - a) \(R \Rightarrow Q\) is **true**.
  - b) \(P \land \neg R\) is **true**.
  - c) \((Q \Rightarrow R) \land R\) is **false**.
  - d) \((Q \land R) \Rightarrow P\) is **true**.

Would you like further clarification or have any questions?

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1. What does it mean when a conditional statement has a contradiction as a consequent?
2. How does the truth value of a disjunction work?
3. Can you explain why a conditional statement with a false antecedent is always true?
4. What are the conditions under which a conjunction is true?
5. How do the truth values of implications depend on the antecedent and consequent?

**Tip:** When solving logic problems, carefully determine the truth values of individual components before combining them into larger expressions.

Explore the truth values of logical statements involving conditional statements and disjunctions. Learn how to analyze complex logic problems with detailed explanations and examples. Suitable for Grades 10-12 students and beyond.

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