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A matroid basis is a matroid independent set that is a maximal element under inclusion of subsets: no additional element of the ground set can be admitted. Maximum cardinality means having the largest size among all matroid independent sets. Maximality tests whether this particular selection can grow; maximum size compares it with every selection. In a matroid the two conditions coincide, so all matroid bases have a common size. They need not have equal weight. ^[ref]

Proof outline. If a maximal matroid independent set $B$ were smaller than another matroid independent set $J$, the matroid independence axioms would admit an element of $J\setminus B$, contradicting maximality. Finiteness allows every matroid independent set to grow to a matroid basis by successive admissible additions. In a general independence system, maximal selections can have different sizes, as in the hereditary nonmatroid greedy obstruction.

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.