Draw a set diagram to determine whether the conclusion follows logically from the premises. If you are the ninth caller, then you win a free ticket. You are not the ninth caller. Therefore, you did not win a free ticket. Valid argument Invalid argument If you have the flu , then you need to stay home. You need to stay home. Therefore, you have the flu. Valid argument Invalid argument

To determine whether the conclusions follow logically from the premises, we can analyze each argument using logical reasoning and set diagrams.<br /><br />### Argument 1:<br />**Premises:**<br />1. If you are the ninth caller, then you win a free ticket.<br />2. You are not the ninth caller.<br /><br />**Conclusion:**<br />- Therefore, you did not win a free ticket.<br /><br />**Analysis:**<br />This argument follows the form of "If P, then Q. Not P. Therefore, not Q." This is known as denying the antecedent, which is a formal fallacy in logic. Just because you are not the ninth caller does not necessarily mean you cannot win a free ticket through other means.<br /><br />**Set Diagram:**<br />- Let Set A represent people who are the ninth caller.<br />- Let Set B represent people who win a free ticket.<br />- The premise states that Set A is a subset of Set B.<br />- However, being outside Set A (not the ninth caller) does not imply being outside Set B (not winning a free ticket).<br /><br />**Conclusion:**<br />- **Invalid argument**<br /><br />### Argument 2:<br />**Premises:**<br />1. If you have the flu, then you need to stay home.<br />2. You need to stay home.<br /><br />**Conclusion:**<br />- Therefore, you have the flu.<br /><br />**Analysis:**<br />This argument follows the form of "If P, then Q. Q. Therefore, P." This is known as affirming the consequent, which is also a formal fallacy in logic. Needing to stay home could be due to reasons other than having the flu.<br /><br />**Set Diagram:**<br />- Let Set C represent people who have the flu.<br />- Let Set D represent people who need to stay home.<br />- The premise states that Set C is a subset of Set D.<br />- Being in Set D (needing to stay home) does not necessarily mean being in Set C (having the flu).<br /><br />**Conclusion:**<br />- **Invalid argument**<br /><br />In both cases, the arguments are invalid because they commit logical fallacies.