A matroid on a finite set consists of a nonempty family of independent subsets closed under taking subsets and satisfying exchange: if independent sets obey , some has independent. Linear independence of vectors is the motivating example. A pregeometry is a finitary closure-theoretic extension to possibly infinite sets.
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A **matroid** is a combinatorial structure that generalizes the notion of linear independence in vector spaces to more abstract settings. It is defined by a pair \((S, I)\), where: - \(S\) is a finite set of elements. - \(I\) is a collection of subsets of \(S\) (called independent sets) that satisfy certain properties.