# Prove that a structure related to 1-tree is a matroid?

I am studying this chapter and I am stuck on page 262. In fact, I do not understand why the pair (E, $$\tilde{\mathcal{S}}$$) is a matroid where $$\tilde{\mathcal{S}}$$ is the set of all the subsets of all 1-trees on the complete graph $$K_n$$. Following the definition in Wikipedia, I am supposed to prove three things:

1. The empty set is in $$\tilde{\mathcal{S}}$$: this is true, as it is a subset of every set, and thus a subset of every 1-tree.
2. Every subset of $$S\in \tilde{\mathcal{S}}$$ is in $$\tilde{\mathcal{S}}$$ as well: this is trivially true, as every subset of a subset of a 1-tree is a subset of a 1-tree as well.
3. If $$I,J\in\tilde{\mathcal{S}}$$ and $$\vert I \vert < \vert J \vert$$, then there exists $$e \in J \setminus I$$ such that $$I\cup{e}$$ is still in $$\tilde{\mathcal{S}}$$: This is where I am stuck. Consider the following examples: Let $$K_5$$ be the complete graph on 5 nodes and consider the 1-tree $${(1, 2), (1, 4), (2, 3), (4, 3), (4,5)}$$ and let $$I = {(1, 2)}$$, and $$J = {(1, 4), (2, 3)}$$. The two sets are both subsets of the original 1-tree, but they are disjoint, so I cannot prove the existence of such $$e$$.

For property 3, $$I$$, $$J$$, and $$I \cup e$$ need not be subsets of the same $$1$$-tree. In your example, you can take either edge in $$J$$.