A set can be pictured as a tree.

If one of A, B, C, D, E in this example is itself a set, we can expand the diagram further. For instance, if A={C, D}, then we get the tree

**Exploration 4.1.**How would you represent a singleton and the empty set as a tree? Using this insight, represent the following sets as trees:

- {{}}
- {{{}}}
- {{},{{}}}
- {{A, {{}, B}}}

Note that representing a set as a tree introduces even more redundant information than in the symbolic notation of sets. First, let's list some redundancies in the symbolic notation:

- Repetition of elements: the set {A} is the same set as the set {A, A} since both sets have exactly one element, the object A (recall that just because we mention a symbol twice does not mean it has two separate meanings).
- Order of elements: the set {A, B} is the same set as the set {B, A} because both sets have the same elements.

With the tree notation, in addition to the redundancies inherited from the symbolic notation, there is also a redundancy of the shape of the drawing, such as the length of branches, for instance. For instance, the last diagram above represents the same set as the following one:

**Exploration 4.2.**Represent as trees each of the quotients of the set X from the example above.