Question

Write each of the following sets in set-builder notation. $$ \\{\ldots,-8,-3,2,7,12,17, \ldots\\} $$

Short answer

The given set in set-builder notation is \({x \in \mathbb{Z} | x = -8 +5n, n \in \mathbb{Z}\}.

Step-by-step solution

Identify the Pattern

Inspect the given set visually. Each successive element appears to be 5 more than the previous one and starts from -8. So, the set can be defined as a list of numbers where each number is 5 more than the previous number.

Derive the Rule

Express this observation in the form of a mathematical rule. Every number in this set can be written as -8 + 5n, where n is some integer. For example, when n = 0, we get -8, when n = 1, we get -3, and so on.

Write in Set-Builder Notation

Translate this rule into set-builder notation. The set-builder form of a set is represented by \({x: property}\) or \({x | property}\), which means 'the set of all x such that this property or condition is satisfied'. Thus, in set-builder notation, this set can be written as \({x \in \mathbb{Z} | x = -8 +5n, n \in \mathbb{Z}\}. This represents the set of all x in the set of integers such that x can be written in the form -8 + 5n, for some integer n.