Question: 1) Use the set-roster notation to indicate the elements in each of the following sets. a) S = {n ∈ Z| n = (−1)^k , for some integer k}. b) T = {m ∈ Z |m = 1 + (−1)^i, for some integer i}. c) U = {r ∈ Z| 2 ≤ r ≤ −2} d) V = {s ∈ Z| s > 2 or s < 3} e) W = {t ∈ Z | 1 < t < −3} f) X = {u ∈ Z | u ≤ 4 or u ≥ 1}
1) Use the set-roster notation to indicate the elements in each of the following sets.

a) S = {n ∈ Z| n = (−1)^k , for some integer k}.

b) T = {m ∈ Z |m = 1 + (−1)^i, for some integer i}.

c) U = {r ∈ Z| 2 ≤ r ≤ −2}

d) V = {s ∈ Z| s > 2 or s < 3}

e) W = {t ∈ Z | 1 < t < −3}

f) X = {u ∈ Z | u ≤ 4 or u ≥ 1}

Let's go through each set one by one and express them in **set-roster notation**, which involves listing out the elements of the set explicitly.

### a) \( S = \{n \in \mathbb{Z} \mid n = (-1)^k, \text{ for some integer } k\} \)

- The expression \( (-1)^k \) alternates between 1 and -1 depending on whether \( k \) is even or odd.
- So, \( S = \{-1, 1\} \).

### b) \( T = \{m \in \mathbb{Z} \mid m = 1 + (-1)^i, \text{ for some integer } i\} \)

- The expression \( (-1)^i \) alternates between 1 and -1 depending on whether \( i \) is even or odd.
- When \( (-1)^i = 1 \), \( m = 1 + 1 = 2 \).
- When \( (-1)^i = -1 \), \( m = 1 + (-1) = 0 \).
- So, \( T = \{0, 2\} \).

### c) \( U = \{r \in \mathbb{Z} \mid 2 \leq r \leq -2\} \)

- There are no integers that satisfy \( 2 \leq r \leq -2 \), as this is logically impossible.
- So, \( U = \emptyset \) (the empty set).

### d) \( V = \{s \in \mathbb{Z} \mid s > 2 \text{ or } s < 3\} \)

- \( s > 2 \) gives integers greater than 2: \( 3, 4, 5, \dots \)
- \( s < 3 \) gives integers less than 3: \( \dots, -2, -1, 0, 1, 2 \)
- Combining these two ranges: \( V = \{\dots, -2, -1, 0, 1, 2, 3, 4, 5, \dots \} \), which is just \( \mathbb{Z} \), the set of all integers.

### e) \( W = \{t \in \mathbb{Z} \mid 1 < t < -3\} \)

- This condition is impossible because no integer can be both greater than 1 and less than -3.
- So, \( W = \emptyset \) (the empty set).

### f) \( X = \{u \in \mathbb{Z} \mid u \leq 4 \text{ or } u \geq 1\} \)

- The condition \( u \leq 4 \text{ or } u \geq 1 \) describes the integers in the range \( u \leq 4 \), because any integer greater than or equal to 1 is also covered by this range.
- So, \( X = \{\dots, -2, -1, 0, 1, 2, 3, 4\} \).

### Final Answers in Set-Roster Notation:
1. \( S = \{-1, 1\} \)
2. \( T = \{0, 2\} \)
3. \( U = \emptyset \)
4. \( V = \mathbb{Z} \)
5. \( W = \emptyset \)
6. \( X = \{\dots, -2, -1, 0, 1, 2, 3, 4\} \)

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Would you like further clarification on any step?  
Here are 5 related questions to deepen your understanding:

1. How do you handle set-builder notation when dealing with inequalities?
2. What is the difference between roster notation and set-builder notation?
3. How would you describe the union of two sets in roster notation?
4. What happens when set conditions contradict each other, as in \( W \)?
5. Can set-roster notation be used for infinite sets? How does that look?

**Tip**: When working with inequalities in set notation, visualize the number line to better understand the set's boundaries.

1) Use the set-roster notation to indicate the elements in each of the following sets. a) S = {n ∈ Z| n = (−1)^k , for some integer k}. b) T = {m ∈ Z |m = 1 + (−1)^i, for some integer i}. c) U = {r ∈ Z| 2 ≤ r ≤ −2} d) V = {s ∈ Z| s > 2 or s < 3} e) W = {t ∈ Z | 1 < t < −3} f) X = {u ∈ Z | u ≤ 4 or u ≥ 1}

Explore how to use set-roster notation to represent integer sets S, T, U, V, W, and X. This problem introduces key set theory concepts and demonstrates how to express integer sets with conditions such as alternating powers of (-1), inequalities, and logical contradictions. Ideal for students in Grades 9-11.

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