5) If (pQ)(SvR) is false, then  the truth value of a)SR is----

                                                                                b)PvS is-----

                                                                                   c)(PQ)R is--------

                                                                                   d)PR is-------


Given that \((p \land Q) \Rightarrow (\neg S \lor R)\) is false, we can determine the truth values of the statements \(S \Rightarrow R\), \(\neg P \lor \neg S\), \((P \land \neg Q) \Leftrightarrow R\), and \(P \land R\) as follows:

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

A conditional statement \(A \Rightarrow B\) is false only when \(A\) is true and \(B\) is false. 

So, for \((p \land Q) \Rightarrow (\neg S \lor R)\) to be false:
- \(p \land Q\) must be true (i.e., \(p\) is true and \(Q\) is true).
- \(\neg S \lor R\) must be false.

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

So, we have:
- \(S\) is true.
- \(R\) is false.
- \(p\) is true.
- \(Q\) is true.

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

a) \(S \Rightarrow R\):

\(S \Rightarrow R\) is true if \(S\) is false or \(R\) is true. Since \(S\) is true and \(R\) is false, the statement \(S \Rightarrow R\) is false.

b) \(\neg P \lor \neg S\):

\(\neg P \lor \neg S\) is true if either \(\neg P\) or \(\neg S\) is true. Given \(P\) is true and \(S\) is true, \(\neg P\) is false and \(\neg S\) is false. Therefore, \(\neg P \lor \neg S\) is false.

c) \((P \land \neg Q) \Leftrightarrow R\):

\(P \land \neg Q\) would be false since \(Q\) is true. \(R\) is false. The equivalence \((P \land \neg Q) \Leftrightarrow R\) will be true since both sides are false.

d) \(P \land R\):

\(P \land R\) is true if both \(P\) and \(R\) are true. Since \(P\) is true and \(R\) is false, \(P \land R\) is false.

### Summary of Truth Values:
a) \(S \Rightarrow R\) is **false**.  
b) \(\neg P \lor \neg S\) is **false**.  
c) \((P \land \neg Q) \Leftrightarrow R\) is **true**.  
d) \(P \land R\) is **false**.

Would you like more details or have any questions?

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1. What is the truth table for a conditional statement?
2. How does the truth value of a conjunction work?
3. Can you explain the negation of a disjunction?
4. What is the biconditional operation in logic?
5. How does one determine the truth value of complex logical statements?

**Tip:** When dealing with logical statements, it's essential to carefully consider the conditions under which compound statements are true or false.

Explore the analysis of logical statements with a focus on truth values. This page discusses the truth values of conditional statements, disjunctions, conjunctions, and equivalence in logic, using a detailed example. Suitable for high school students studying logic.

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