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 ∃y∀x, enrolled(x, y), where x is a student at champlain college and y is a degree ∃x∀y, enrolled(x, y) ∃y∀x, enrolled(x, y), where x is a student at champlain college and y is a degree none of the alternatives is correct all of the alternatives are correct

1) ∃x∃y, ¬enrolled(x, y)
2) ∀x∃y, ¬enrolled(x, y)
3) ∃y∀x, ¬enrolled(x, y)
4) ∃x∀y, ¬enrolled(x, y)

Answer 1

The correct negation of the statement ∀x∀y, enrolled(x, y) is ∃x∃y, ¬enrolled(x, y), meaning there is at least one student not enrolled in at least one degree group.

Step-by-step explanation:

To negate the statement ∀x∀y, enrolled(x, y), where 'x' is a student at Champlain College and 'y' is a degree group, we look for a proposition that states there is at least one student and at least one degree group where the student is not enrolled. The correct negation is the existence of at least one counterexample to the universal claim. Thus, the correct negated proposition is ∃x∃y, ¬enrolled(x, y), which means 'There exists at least one student x at Champlain College and at least one degree group y such that x is not enrolled in y'.

Understanding valid deductive inferences like disjunctive syllogism, modus ponens, and modus tollens is essential in the negation of propositions. In negating the statement, we take the universal affirmative and provide a counterexample, creating a universal negative statement. Negating quantified statements often involves switching from universal (∀) to existential (∃) quantifiers and negating the predicate.