Question

For the following formulas/propositions, indicate the option that corresponds to negating it: ∀x∃y, enrolled(x, y), where x is a student at champlain college and y is a degree group of answer choices ∃x∀y, enrolled(x, y), where x is a student at champlain college and y is a degree ∀x∃y, enrolled(x, y), where x is a student at champlain college and y is a degree none of the alternatives is correct?

1) ∃x∀y, enrolled(x, y), where x is a student at champlain college and y is not a degree
2) ∀x∃y, enrolled(x, y), where x is not a student at champlain college and y is a degree
3) ∀x∃y, not enrolled(x, y), where x is a student at champlain college and y is a degree
4) ∃x∀y, not enrolled(x, y), where x is a student at champlain college and y is a degree

Answer 1

The correct negation of the proposition '∀x∃y, enrolled(x, y)' is option 4) '∃x∀y, not enrolled(x, y)', implying that there is at least one student at Champlain College not enrolled in any degree.

Step-by-step explanation:

To negate the proposition ∀x∃y, enrolled(x, y), where x is a student at Champlain College and y is a degree, you want to find the option that expresses that there exists at least one student who is not enrolled in any degree. The correct negation is to say 'There exists a student such that for all degree groups, the student is not enrolled in that degree group.' Therefore, the correct negation would be option 4) ∃x∀y, not enrolled(x, y), where x is a student at Champlain College and y is a degree. This states that there is at least one student at Champlain College who is not enrolled in any degree, which is the logical opposite of saying that every student is enrolled in some degree.