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.