Definition:Matroid/Definition 2

Definition
Let $S$ be a finite set.

Let $\mathscr I$ be a independence system.

The ordered pair $M = \struct{S, \mathscr I}$ is called a matroid on $S$, or simply a matroid when the context is obvious, if $\mathscr I$ also satisfies:

Matroid Axioms
The properties of a matroid are as follows.

For a given matroid $M = \struct{S, \mathscr I}$ these statements hold true: