Key Concepts and Formulas

• Propositional Logic: Understanding the truth values (True/False) of propositions and how they combine using logical operators (∨ - OR, ∧ - AND, ∼ - NOT).
• Truth Tables: Although not explicitly constructed here, the underlying principle is to find a truth assignment that satisfies the maximum number of given compound propositions.
• Satisfiability: The problem revolves around finding a truth assignment to variables that makes a set of compound propositions simultaneously true.

Step-by-Step Solution

Step 1: Analyze the given propositions and identify a common element. We are given the following nine compound propositions:

1. $p \vee r \vee s$
2. $p \vee r \vee \sim s$
3. $p \vee \sim q \vee s$
4. $\sim p \vee \sim r \vee s$
5. $\sim p \vee \sim r \vee \sim s$
6. $\sim p \vee q \vee \sim s$
7. $q \vee r \vee \sim s$
8. $q \vee \sim r \vee \sim s$
9. $\sim p \vee \sim q \vee \sim s$

Notice that $\sim s$ appears in propositions 2, 5, 6, 7, 8, and 9. If we set $s$ to False, i.e., $s = F$, these six propositions will become true regardless of the values of $p, q,$ and $r$.

Step 2: Assign a truth value to 's' to maximize the number of initially satisfied propositions. Let's assign $s = F$. This makes the following propositions true: 2. $p \vee r \vee \sim s \equiv p \vee r \vee T \equiv T$ 3. $\sim p \vee \sim r \vee \sim s \equiv \sim p \vee \sim r \vee T \equiv T$ 4. $\sim p \vee q \vee \sim s \equiv \sim p \vee q \vee T \equiv T$ 5. $q \vee r \vee \sim s \equiv q \vee r \vee T \equiv T$ 6. $q \vee \sim r \vee \sim s \equiv q \vee \sim r \vee T \equiv T$ 7. $\sim p \vee \sim q \vee \sim s \equiv \sim p \vee \sim q \vee T \equiv T$

Now we need to check if we can make the remaining propositions true as well.

Step 3: Analyze the remaining propositions with s = F. The remaining propositions are:

1. $p \vee r \vee s \equiv p \vee r \vee F \equiv p \vee r$
2. $p \vee \sim q \vee s \equiv p \vee \sim q \vee F \equiv p \vee \sim q$
3. $\sim p \vee \sim r \vee s \equiv \sim p \vee \sim r \vee F \equiv \sim p \vee \sim r$

Step 4: Find values for p, q, and r to maximize the number of remaining true propositions. We want to make $p \vee r$, $p \vee \sim q$, and $\sim p \vee \sim r$ true.

Let's consider the case where $p = T$ and $r = T$. Then:

• $p \vee r \equiv T \vee T \equiv T$
• $p \vee \sim q \equiv T \vee \sim q \equiv T$
• $\sim p \vee \sim r \equiv F \vee F \equiv F$

This doesn't work since $\sim p \vee \sim r$ is false.

Let's try $p = T$ and $r = F$. Then:

• $p \vee r \equiv T \vee F \equiv T$
• $p \vee \sim q \equiv T \vee \sim q \equiv T$
• $\sim p \vee \sim r \equiv F \vee T \equiv T$

This works! In this case, we can choose any value for $q$, and all three remaining propositions will be true.

Step 5: Determine the maximum number of true propositions. If we set $s = F$, $p = T$, and $r = F$, the nine propositions become:

1. $p \vee r \vee s \equiv T \vee F \vee F \equiv T$
2. $p \vee r \vee \sim s \equiv T \vee F \vee T \equiv T$
3. $p \vee \sim q \vee s \equiv T \vee \sim q \vee F \equiv T$ (since $T \vee \sim q$ is always true)
4. $\sim p \vee \sim r \vee s \equiv F \vee T \vee F \equiv T$
5. $\sim p \vee \sim r \vee \sim s \equiv F \vee T \vee T \equiv T$
6. $\sim p \vee q \vee \sim s \equiv F \vee q \vee T \equiv T$
7. $q \vee r \vee \sim s \equiv q \vee F \vee T \equiv T$
8. $q \vee \sim r \vee \sim s \equiv q \vee T \vee T \equiv T$
9. $\sim p \vee \sim q \vee \sim s \equiv F \vee \sim q \vee T \equiv T$

Therefore, all 9 propositions can be made true simultaneously.

Common Mistakes & Tips

• Overlooking Simplifications: Always simplify the propositions after assigning truth values to see if further simplification is possible.
• Trying All Combinations: It is not necessary to check all $2^4 = 16$ possible truth assignments. Start by looking for patterns that can satisfy multiple propositions simultaneously.
• Misunderstanding the OR operator: Remember that $A \vee B$ is true if either $A$ is true, $B$ is true, or both are true.

Summary

We analyzed the given compound propositions and identified a variable ($\sim s$) that appeared frequently. By setting $s$ to False, we made six propositions automatically true. Then, we found values for $p$ and $r$ that made the remaining three propositions also true. Therefore, all nine propositions can be made simultaneously true by a suitable assignment of truth values.

Final Answer The final answer is \boxed{9}.