To express the set $S = \{10, 14, 18, 28, 30, 35\}$ in builder notation, we need to identify a pattern or condition that applies to each element in $S$.

It seems these numbers increase by various increments but can be represented as a subset of integers within a specific range.

Here's a possible builder notation for the set $S$:

$S = \{ x \in \mathbb{Z} \mid 10 \leq x \leq 35 \text{ and } x \equiv 0 \pmod{4} \text{ or } x \equiv 2 \pmod{5} \}$

This notation describes a set $S$ of integers $x$ that are between 10 and 35, inclusive, and meet the modulus conditions, capturing the specific values in $S$.

Would you like further clarification, or do you have more questions on this topic?

Here are five related questions for further practice:

1. How would you write the set $\{2, 4, 6, 8, 10\}$ in builder notation form?
2. What does builder notation mean in set theory?
3. How can you use builder notation to define sets with a specific arithmetic progression?
4. Can you write the set of all even numbers less than 20 in builder notation?
5. How would you represent the set of all prime numbers less than 50 in builder notation?

Tip: When using builder notation, always check for any common property among elements, such as arithmetic progressions or divisibility, to simplify your expression.