of these is the set-builder notation of the set \( \{-10,-5,0,5,10\} \)

The set-builder notation for the set \(\{-10, -5, 0, 5, 10\}\) can be expressed as \(\{ x \in \mathbb{Z} \mid x = 5n, n \in \{-2, -1, 0, 1, 2\} \}\), which captures the essence of how these numbers are formed. It succinctly indicates that \(x\) is an integer that can be represented as \(5n\), where \(n\) takes values from a defined set. This set is a great example of how we can use rules to define a collection of numbers. In practical applications, set-builder notation is commonly used in mathematics to define complex sets crisply. It helps in computer programming, database querying, and even probability theory where you can succinctly specify criteria for data collections without listing every element.