Let's go through the sets one by one and list all their members.

1. Set A: $A = \{x \in \mathbb{Z} : 0 \leq x < 7\}$

   • This set includes all integers $x$ such that $0 \leq x < 7$.
   • The members of set $A$ are: $\{0, 1, 2, 3, 4, 5, 6\}$.

2. Set B: $B = \{x \in \mathbb{Z} : -1 < x \leq 2\}$

   • This set includes all integers $x$ such that $-1 < x \leq 2$.
   • The members of set $B$ are: $\{0, 1, 2\}$.

3. Set C: $C = \{x \in \mathbb{Z} : x = -(-1)^n + 1 \text{ for some integer } n\}$

   • This set involves the formula $x = -(-1)^n + 1$ where $n$ is an integer. We can find the values of $x$ by substituting values of $n$:
     • When $n = 0$: $x = -(-1)^0 + 1 = -1 + 1 = 0$
     • When $n = 1$: $x = -(-1)^1 + 1 = 1 + 1 = 2$
     • When $n = 2$: $x = -(-1)^2 + 1 = -1 + 1 = 0$ (Repeats, so cycle continues)
   • Therefore, the members of set $C$ are: $\{0, 2\}$.

4. Set D: $D = \{x \in \mathbb{Z} : -2 < x < 4\}$

   • This set includes all integers $x$ such that $-2 < x < 4$.
   • The members of set $D$ are: $\{-1, 0, 1, 2, 3\}$.

Let me know if you need further details or explanations. Here are some related questions you might want to explore:

1. How can you determine the range of a set if given different inequalities?
2. What happens if you change the boundaries in the set definitions (e.g., from closed intervals to open intervals)?
3. How would set $C$ change if you adjusted the formula slightly?
4. Can you represent any of these sets using a union or intersection of other sets?
5. What are the general properties of the integers that belong to set $A$?

Tip: When dealing with set definitions involving formulas, start by plugging in small integer values to identify patterns or cycles.