Writing All Subsets of a Set

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

Understanding how to write all subsets of a given set is a fundamental skill in set theory, which is a cornerstone of real analysis. The process involves considering all possible combinations of elements within the set, including the empty set and the set itself. The power set of any set is the collection of all these subsets. It inherently has a higher cardinality than the original set, specifically, if the original set has n elements, its power set will have $2^n$ elements due to the binary nature of choosing to include or exclude each element.

The practical computation of all subsets can be approached systematically by starting with smaller subsets and incrementally building to larger ones, making sure each potential subset is accounted for. This method often involves recursive thinking and can be linked to binary logic, where each element has a state of being included or not included.

This problem also invites reflection on larger concepts such as the idea of cardinality, which discusses the size or number of elements in a set and relates deeply to understanding the infinite in real analysis. Real, rational, and natural numbers are frequently discussed in these contexts, as they form the foundational number sets in mathematics.