The rank of a subset S of a matroid is the size of a maximal independent subset of S. The map 2^E $\to \mathbb{N}$, S $\mapsto$ rank(S), is called the rank function, and completely determines the matroid.
i1 : M = matroid({a,b,c,d},{{a,b},{a,c}})
o1 = a matroid of rank 2 on 4 elements
o1 : Matroid
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i2 : for s in subsets M_* do print(toString s | " has rank " | rank_M s)
{} has rank 0
{a} has rank 1
{b} has rank 1
{a, b} has rank 2
{c} has rank 1
{a, c} has rank 2
{b, c} has rank 1
{a, b, c} has rank 2
{d} has rank 0
{a, d} has rank 1
{b, d} has rank 1
{a, b, d} has rank 2
{c, d} has rank 1
{a, c, d} has rank 2
{b, c, d} has rank 1
{a, b, c, d} has rank 2
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