= Matroid independence axioms
{title2=$M=(E,\mathcal I)$}
Let $E$ be a <finite set> and $\mathcal I\subseteq\mathcal P(E)$ a family of <subsets>, where $\mathcal P(E)$ is the <power set>. The <matroid independence axioms> are
$$
\begin{aligned}
&\varnothing\in\mathcal I,\\
&I\in\mathcal I,\ J\subseteq I\quad\Longrightarrow\quad J\in\mathcal I,\\
&I,J\in\mathcal I,\ |I|<|J|\quad\Longrightarrow\quad
\exists e\in J\setminus I:\ I\cup\{e\}\in\mathcal I.
\end{aligned}
$$
The second condition is heredity; the third is exchange, also called augmentation. The <set difference> $J\setminus I$ matters: the new element must actually increase the <cardinality> of $I$. Exchange adds one element without requiring any deletion from $I$. Nonemptiness together with heredity is equivalent to the first condition. <Matroid optimization by Michel Goemans>[Michel X. Goemans, Matroid optimization, section 5.1, p. 1, axioms (I1) and (I2)]{ref}
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