Question

List all the subsets of the following sets. $$ \\{\\{0,1\\},\\{0,1,\\{2\\}\\},\\{0\\}\\} $$

Short answer

The subsets of the given set are: {}, \{\{0,1\}\}, \{\{0,1,\{2\}\}\}, \{\{0\}\}, \{\{0,1\}, \{0,1,\{2\}\}\}, \{\{0,1\}, \{0\}\}, \{\{0,1,\{2\}\}, \{0\}\}, and \{\{0,1\},\{0,1,\{2\}\},\{0\}\}

Step-by-step solution

Identify the elements of the set

The given set has three elements: \{0,1\}, \{0,1,\{2\}\}, and \{0\}.

List subsets with no elements

The empty set, denoted by {}, is a subset of every set, so it is a subset in this case as well.

List subsets with one element

The subsets with one element are the sets that contain only one of the elements of the original set. They are: \{\{0,1\}\}, \{\{0,1,\{2\}\}\}, and \{\{0\}\}

List subsets with two elements

These subsets contain any two of the elements of the original set. They are: \{\{0,1\}, \{0,1,\{2\}\}\}, \{\{0,1\}, \{0\}\}, and \{\{0,1,\{2\}\}, \{0\}\}

List subsets with three elements

This subset contains all the elements of the original set, which is the set itself: \{\{0,1\},\{0,1,\{2\}\},\{0\}\}