To solve the problem of finding the maximum number of compound propositions that can be made simultaneously true by assigning truth values to \( p, q, r, \) and \( s \), we will analyze each proposition step by step.
### Given Propositions:
1. \( p \lor r \lor s \)
2. \( p \lor r \lor \neg s \)
3. \( p \lor \neg q \lor s \)
4. \( \neg p \lor \neg r \lor s \)
5. \( \neg p \lor \neg r \lor \neg s \)
6. \( \neg p \lor q \lor \neg s \)
7. \( q \lor \neg q \lor \neg s \)
8. \( q \lor r \lor \neg s \)
9. \( q \lor \neg r \lor \neg s \)
10. \( \neg p \lor \neg q \lor \neg s \)
### Step-by-Step Solution:
**Step 1: Analyze the propositions involving \( \neg s \)**
We first note that several propositions contain \( \neg s \):
- \( p \lor r \lor \neg s \) (2)
- \( p \lor \neg q \lor \neg s \) (3)
- \( \neg p \lor \neg r \lor \neg s \) (5)
- \( \neg p \lor q \lor \neg s \) (6)
- \( q \lor \neg q \lor \neg s \) (7)
- \( q \lor r \lor \neg s \) (8)
- \( q \lor \neg r \lor \neg s \) (9)
- \( \neg p \lor \neg q \lor \neg s \) (10)
If we set \( s = \text{False} \), then \( \neg s = \text{True} \). This means all propositions that include \( \neg s \) will be true.
**Step 2: Count the propositions that can be true with \( s = \text{False} \)**
By setting \( s = \text{False} \):
- Propositions 2, 3, 5, 6, 7, 8, 9, and 10 are true, which gives us **8 true propositions**.
**Step 3: Analyze the remaining propositions without \( \neg s \)**
Now we need to check the remaining propositions:
1. \( p \lor r \lor s \) (1) - This will be false since \( s = \text{False} \).
2. \( p \lor r \lor \neg s \) (2) - This is true since \( \neg s = \text{True} \).
**Step 4: Determine if we can make the remaining propositions true**
To maximize the number of true propositions, we need to check if we can set values for \( p \) and \( r \) such that:
- \( p \lor r \lor s \) is true.
Since \( s = \text{False} \), we need \( p \lor r \) to be true. We can set \( p = \text{True} \) or \( r = \text{True} \).
**Step 5: Conclusion**
With \( s = \text{False} \), \( p = \text{True} \), and \( r = \text{True} \), we can satisfy all propositions:
- \( p \lor r \lor s \) becomes true.
- All other propositions involving \( \neg s \) remain true.
Thus, the maximum number of compound propositions that can be made simultaneously true is **9**.
### Final Answer:
The maximum number of compound propositions that can be made simultaneously true is **9**.
