Subsets of a Set with Nested Sets

Write all subsets of the set $G$, given $G = \{ \mathbb{R}, \{ \mathbb{Q}, \mathbb{N} \} \}$.

The problem of finding all subsets of a set that includes nested sets is a fundamental exercise in understanding how sets can be constructed and deconstructed. This concept is essential in real analysis and forms the basis for more advanced topics such as cardinality and set theory. When dealing with sets, it is important to remember that each element of the power set (or set of all subsets) represents a combination of elements from the original set. This includes the empty set and the set itself as trivial subsets.

In this specific problem, we have a set G that contains real numbers and nested sets as its elements. Understanding and writing down all subsets involves considering subsets that include each element individually, each possible pair of elements, and so on, including the empty set. It helps illuminate the concept of power sets, emphasizing how elements are combined, and the hierarchical nature of sets when they contain other sets. Exploring problems like these help build a solid foundation for topics such as topology and functions, where understanding structure deeply is crucial.

Working through this problem also provides practice in systematically organizing and writing out possibilities, which is not only essential for set theory but also critical for logical reasoning in mathematics broadly. Such exercises can enhance one's ability to approach complex problems methodically, an invaluable skill in advanced mathematics.