Question

List all the subsets of the following sets. $$ \\{\\{\mathbb{R}\\}\\} $$

Short answer

The subsets of the set \{\{\mathbb{R}\}\} are: {}, \{\{\mathbb{R}\}\}

Step-by-step solution

Understand what is a set and a subset

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. A subset of a set A is a set, every element of which is an element of A. For example, if we have a set A = {1, 2, 3}, the subsets are {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, and {1, 2, 3}.

Understand what is a singleton set

A singleton set is a set with only one item. It's denoted as \{a\}, where a is the only element of the set. In our case, our set is \{\{\mathbb{R}\}\}, which is a singleton set because it contains only one set \{\mathbb{R}\}.

List all subsets of the singleton set

For any set, there are always 2 subsets which exist, the empty set and the set itself. Since our set is a singleton set containing the set of real numbers, so it's two subsets are the empty set {} and the set itself \{\{\mathbb{R}\}\}.