Use set-builder notation to describe each set. See Example 2. (More than one description is possible.) {4, 8, 12, 16,...}

Verified step by step guidance

1. Identify the pattern in the given set: {4, 8, 12, 16, ...}. Notice that each element increases by 4, which suggests the set consists of multiples of 4.
2. Express the elements in terms of a variable, say \( n \), where \( n \) represents natural numbers starting from 1. Each element can be written as \( 4n \).
3. Write the set in set-builder notation by specifying the variable and the condition it must satisfy. For example, \( \{ x \mid x = 4n, n \in \mathbb{N} \} \), where \( \mathbb{N} \) denotes the set of natural numbers.
4. Alternatively, you can describe the set by stating that \( x \) is a positive integer multiple of 4, which can be written as \( \{ x \mid x \text{ is a multiple of } 4, x > 0 \} \).
5. Confirm that your set-builder notation accurately represents all elements in the original set and excludes any elements not in the set.

Key Concepts

Set-Builder Notation

Set-builder notation is a concise way to describe a set by specifying a property that its members satisfy. Instead of listing elements, it defines the set as all elements x such that a condition P(x) holds true, e.g., {x | P(x)}. This notation is useful for describing infinite or large sets.

Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. For example, the set {4, 8, 12, 16,...} has a common difference of 4. Recognizing this pattern helps express the set using a formula for the nth term, such as a_n = 4n.

Domain and Variable Constraints in Set Definitions

When using set-builder notation, it is important to specify the domain and constraints on the variable, such as integers or natural numbers. For example, defining the set {4, 8, 12, 16,...} requires stating that n is a natural number to ensure only positive multiples of 4 are included.