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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 $e$.
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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 $J$.

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