= Solution
A <matroid> is a pair $(X,\mathcal I)$, where $X$ is a finite <ground set> and $\mathcal I\subseteq2^X$ is its collection of <matroid independent sets>, satisfying the <matroid independence axioms>:
* $\varnothing\in\mathcal I$.
* If $I\in\mathcal I$ and $J\subseteq I$, then $J\in\mathcal I$.
* If $I,J\in\mathcal I$ and $|I|<|J|$, some $x\in J\setminus I$ satisfies $I\cup\{x\}\in\mathcal I$.
The last axiom is augmentation. In particular, every maximal <matroid independent set> within a given <subset> has maximum <cardinality> there: otherwise augmentation would enlarge it.
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