Final Answer:
Converse: If you enjoy mathematics, then you do math problems every night.
Contrapositive: If you don't enjoy mathematics, then you don't do math problems every night.
Inverse: If you don't do math problems every night, then you don't enjoy mathematics.
Neither: You enjoy mathematics if you do math problems every night.
Neither: You do math problems every night but you don't enjoy mathematics.
Step-by-step explanation:
In logic, the converse of a statement switches the hypothesis and the conclusion. The contrapositive of a statement switches and negates both the hypothesis and the conclusion. The inverse of a statement negates both the hypothesis and the conclusion.
Applying these definitions to the given statements:
1. Converse: "If you enjoy mathematics, then you do math problems every night." - It's the statement where the hypothesis and conclusion are switched.
2. Contrapositive: "If you don't enjoy mathematics, then you don't do math problems every night." - This statement switches and negates both the hypothesis and conclusion.
3. Inverse: "If you don't do math problems every night, then you don't enjoy mathematics." - Here, both the hypothesis and conclusion are negated.
4. Neither: "You enjoy mathematics if you do math problems every night." - This statement does not represent the converse, contrapositive, or inverse.
5. Neither: "You do math problems every night but you don't enjoy mathematics." - This statement doesn't fit the criteria of the converse, contrapositive, or inverse either.
Understanding the relationships between logical statements and their converses, contrapositives, and inverses is crucial in logic and argumentation, aiding in comprehending and formulating valid deductions based on given premises.