Which of the following sentences is logically equivalent to  ¬∃x∃y(P(x,y)∧Q(x,y))
 ?

Question 1Answer

∃x∃y(¬P(x,y)∧¬Q(x,y))
 


∃x∃y(¬P(x,y)∨¬Q(x,y))
 


∀x∀y(¬P(x,y)∧¬Q(x,y))
 


∀x∀y(¬P(x,y)∨¬Q(x,y))

To determine which sentence is logically equivalent to the given expression \( \neg \exists x \exists y (P(x, y) \land Q(x, y)) \), we need to apply logical transformations such as De Morgan's law.

### Step-by-Step Solution:

1. **Start with the given expression:**
   \[
   \neg \exists x \exists y (P(x, y) \land Q(x, y))
   \]

2. **Apply De Morgan's Law:**
   De Morgan's law states that the negation of an existential quantifier can be rewritten as a universal quantifier, and the negation of a conjunction can be rewritten as a disjunction. Applying De Morgan’s law to the above expression:

   \[
   \neg \exists x \exists y (P(x, y) \land Q(x, y)) = \forall x \forall y \neg (P(x, y) \land Q(x, y))
   \]

3. **Distribute the negation inside the parentheses:**
   Again, using De Morgan's law, the negation of a conjunction \( (P(x, y) \land Q(x, y)) \) becomes a disjunction:

   \[
   \forall x \forall y (\neg P(x, y) \lor \neg Q(x, y))
   \]

   So, the given expression is logically equivalent to:
   \[
   \forall x \forall y (\neg P(x, y) \lor \neg Q(x, y))
   \]

### Conclusion:
The sentence that is logically equivalent to \( \neg \exists x \exists y (P(x, y) \land Q(x, y)) \) is:
\[
\boxed{\forall x \forall y (\neg P(x, y) \lor \neg Q(x, y))}
\]

Would you like further details or have any questions? Here are some related questions:

1. What is the equivalent of \( \neg \forall x \forall y (P(x, y) \lor Q(x, y)) \)?
2. How does De Morgan's law apply to quantifiers in general?
3. What is the equivalent form of \( \neg \exists x \forall y (P(x, y)) \)?
4. How would the result change if we had \( \exists x \forall y (P(x, y) \land Q(x, y)) \)?
5. How can these logical equivalences be used in proofs involving negation?

**Tip:** When negating expressions involving quantifiers, De Morgan's laws are key for converting between existential and universal quantifiers.

Understand the logical equivalence of ¬∃x∃y(P(x,y)∧Q(x,y)) to ∀x∀y(¬P(x,y)∨¬Q(x,y)) with a detailed step-by-step solution. Learn about quantifiers, De Morgan's laws, and logical transformations. Suitable for advanced high school students and anyone interested in logic.

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