To solve this task, we need to find the negation of each statement. I'll work through each one using logical negation rules.

Problem

Given a predicate $Z(x, y)$, find the negation of each statement.

Solution

a) $\forall x \forall y : Z(x, y)$

The negation of this is: $\neg (\forall x \forall y : Z(x, y)) \equiv \exists x \exists y : \neg Z(x, y)$ This means that there exists an $x$ and a $y$ such that $Z(x, y)$ is not true.

b) $\forall x \exists y : Z(x, y)$

The negation of this is: $\neg (\forall x \exists y : Z(x, y)) \equiv \exists x \forall y : \neg Z(x, y)$ This means that there exists an $x$ such that for all $y$, $Z(x, y)$ is not true.

c) $\exists x \forall y : Z(x, y)$

The negation of this is: $\neg (\exists x \forall y : Z(x, y)) \equiv \forall x \exists y : \neg Z(x, y)$ This means that for every $x$, there exists a $y$ such that $Z(x, y)$ is not true.

d) $\exists x \exists y : Z(x, y)$

The negation of this is: $\neg (\exists x \exists y : Z(x, y)) \equiv \forall x \forall y : \neg Z(x, y)$ This means that for all $x$ and $y$, $Z(x, y)$ is not true.

Summary of Negations

1. a) $\exists x \exists y : \neg Z(x, y)$
2. b) $\exists x \forall y : \neg Z(x, y)$
3. c) $\forall x \exists y : \neg Z(x, y)$
4. d) $\forall x \forall y : \neg Z(x, y)$

Would you like further explanation on any of these steps?

Here are some related questions:

1. How do we determine the negation of a statement with nested quantifiers?
2. What is the difference between universal and existential quantifiers?
3. How do logical negations apply to combinations of quantifiers and predicates?
4. What are some practical applications of negating quantified statements?
5. How do these rules apply in programming languages with formal logic?

Tip: When negating statements with quantifiers, remember to switch $\forall$ to $\exists$ and vice versa, and apply negation to the predicate itself.