Independence system
= Independence system
{title2=$(E,\mathcal I)$}
An <independence system> consists of a <finite set> $E$ and a nonempty family $\mathcal I$ of <subsets> such that $I\in\mathcal I$ and $J\subseteq I$ imply $J\in\mathcal I$. This is the hereditary condition: deleting chosen elements keeps a selection admissible. In particular, the <empty set> belongs to $\mathcal I$. An <independence system> becomes a <matroid> when it also satisfies the <matroid independence axioms> for exchange. The <hereditary nonmatroid greedy obstruction> shows why deletion alone does not justify a <greedy algorithm>.