**Set-builder notation** is a mathematical notation used to define a set by specifying a property that its members must satisfy. It allows for a concise way to describe sets without having to list all the individual elements.

### General Form
The general form of set-builder notation is:
\[
\{ x \mid \text{property of } x \}
\]
- The curly braces `{}` denote a set.
- The vertical bar `|` (or sometimes `:`) means "such that".
- `x` represents an element of the set.
- The phrase after the bar specifies a condition that elements must satisfy to be included in the set.

### Example
To illustrate, consider the set of all even numbers. In set-builder notation, this could be written as:
\[
\{ x \in \mathbb{Z} \mid x \text{ is even} \}
\]
This reads as "the set of all `x` in the set of integers (`\mathbb{Z}`) such that `x` is even."

### More Complex Example
A more specific example might be the set of all integers between 1 and 10, inclusive. This can be written in set-builder notation as:
\[
\{ x \in \mathbb{Z} \mid 1 \leq x \leq 10 \}
\]
This reads as "the set of all `x` in the integers such that `x` is between 1 and 10, inclusive."

### Applications of Set-Builder Notation
Set-builder notation is particularly useful when dealing with infinite sets, or when a set is defined by a specific rule or condition rather than a simple list of elements.

### Key Points:
1. **Curly braces `{}`** indicate that you are defining a set.
2. **Variable `x`** represents an element of the set.
3. **Condition after `|` (or `:`)** defines the rule that elements must satisfy to be included in the set.

Would you like more examples or have any questions?

### 5 Related Questions to Expand Understanding:
1. How do you use set-builder notation to represent the set of all prime numbers?
2. Can set-builder notation be used to define sets in higher dimensions (e.g., subsets of the plane or space)?
3. What is the difference between set-builder notation and roster notation?
4. How do you express an empty set using set-builder notation?
5. How does set-builder notation apply to defining intervals on the real number line?

### Tip:
When using set-builder notation, ensure the property is clearly defined to avoid ambiguity in which elements belong to the set.

Explore set-builder notation with clear definitions and examples. Understand how to define sets using concise mathematical rules, avoiding the need to list individual elements. Suitable for Grades 9-12.

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