matroid independence axioms

Hassler Whitney’s definition of a matroid was based upon the idea of independent sets and was given in terms of the following three axioms:

A finite set $E$ along with a collection $ℐ$ of subsets of $E$ is a matroid if $M=(E,ℐ)$ meets the following criteria:
(I1) $\mathrm{\varnothing }\in ℐ$;
(I2) If $I_1\in ℐ$ and $I_2\subseteq I_1$, then $I_2\in ℐ$;
(I3) If $I_1,I_2\in ℐ$ with , then there is some $e\in I_2$ such that $I_1\cup \{e\}\in ℐ$.

The third axiom, (I3), is equivalent to the following alternative axiom:

(I3*) If $S,T\in ℐ$ and $S,T\subset U\subset E$ and $S$ and $T$ are both maximal subsets of $U$ with the property that they are in $ℐ$, then $|S|=|T|$.

Proof.

Suppose (I3) holds, and that $S$ and $T$ are maximal independent subsets of $A\subseteq E$. Also assume, without loss of generality, that . Then there is some $e\in T$ such that $S\cup \{e\}\in ℐ$, but $S\subset S\cup \{e\}\subseteq A$, contradicting maximality of $S$.
Now suppose that (I3*) holds, and assume that $S,T\in ℐ$ with . Let $A=S\cup T$. Then $S$ cannot be maximal in $A$ by (I3*), so there must be elements $e_i\in A$ such that $S\cup \{e_i\}\in ℐ$ is maximal, and by construction these $e_i\in T$. So (I3) holds.∎