Basis of a matroid ($B$)

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.