The set $A = \{1, 3, 5, \dots \}$ represents the set of all odd positive integers. To express this set using set-builder notation, you describe the elements of the set in terms of a defining property. The set $A$ can be described as follows:

$A = \{ x \mid x = 2n - 1, n \in \mathbb{N} \}$

Explanation:

• $x$ represents the elements of the set.
• $x = 2n - 1$ describes the general form of odd integers.
• $n \in \mathbb{N}$ means that $n$ is a natural number (positive integers starting from 1).

This set-builder notation states that $A$ contains all numbers that can be expressed as $2n - 1$ where $n$ is a natural number.

Do you want further clarification on any part of this?

Here are some questions for practice:

1. How would you describe the set of even numbers using set-builder notation?
2. Can you describe the set of all multiples of 3 using set-builder notation?
3. How would you represent the set of perfect squares using set-builder notation?
4. How would you describe the set $B = \{2, 4, 6, \dots \}$ using set-builder notation?
5. Can you represent the set of prime numbers using set-builder notation?

Tip: Set-builder notation is a powerful way to describe sets concisely using logical conditions or equations.