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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.

Write down all the members of the following sets: (i) A = {x ∈ Z : 0 ≤ x < 7}, (ii) B = {x ∈ Z : -1 < x ≤ 2}, (iii) C = {x ∈ Z : x = -(-1)^n + 1 for some integer n}, (iv) D = {x ∈ Z : -2 < x < 4}

This solution explores set theory and integer membership with inequalities. Learn how to find members of sets A, B, C, and D based on specific conditions, involving sequences and inequalities. Suitable for Grades 6-9.

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