1. **State the problem:** We are given two statements: - $p$: A month begins with the letter M. - $q$: It has 31 days. We need to find the conditional, inverse, converse, and contrapositive statements of $p \to q$ and their truth values. 2. **Conditional statement:** The conditional is $p \to q$: "If a month begins with M, then it has 31 days." 3. **Truth value of conditional:** Months beginning with M are March and May. - March has 31 days (true). - May has 31 days (true). So, the conditional is **true**. 4. **Inverse statement:** The inverse is $\neg p \to \neg q$: "If a month does not begin with M, then it does not have 31 days." 5. **Truth value of inverse:** Months not beginning with M but having 31 days include January, July, August, October, December. So, the inverse is **false**. 6. **Converse statement:** The converse is $q \to p$: "If a month has 31 days, then it begins with M." 7. **Truth value of converse:** Months with 31 days include January, March, May, July, August, October, December. Only March and May begin with M. So, the converse is **false**. 8. **Contrapositive statement:** The contrapositive is $\neg q \to \neg p$: "If a month does not have 31 days, then it does not begin with M." 9. **Truth value of contrapositive:** Months without 31 days are February, April, June, September, November. None of these begin with M. So, the contrapositive is **true**. 10. **Symbolic notation for inverse:** The inverse of $p \to q$ is $\neg p \to \neg q$, which corresponds to option D. 11. **True bi-conditional statement:** Angles are supplementary if and only if their sum is 180° is true (option B). 12. **Law used or invalid conclusion:** - Given: If you live in Orlando, then you live in Florida. - Morgan does not live in Orlando. - Conclusion: Morgan does not live in Florida. This is an invalid conclusion (denying the hypothesis). 13. **Law used or invalid conclusion:** - Given: If two angles in a triangle are 40° and 30°, then the third angle measures 110°. - Given: If a triangle has an angle that measures 110°, then it is an obtuse triangle. - Conclusion: If two angles in a triangle are 40° and 30°, then it is an obtuse triangle. This uses the Law of Syllogism. Final answers: 7. Conditional: If a month begins with M, then it has 31 days. Truth Value: True 8. Inverse: If a month does not begin with M, then it does not have 31 days. Truth Value: False 9. Converse: If a month has 31 days, then it begins with M. Truth Value: False 10. Contrapositive: If a month does not have 31 days, then it does not begin with M. Truth Value: True 11. D 12. B 13. Invalid 14. Law of Syllogism