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.