Past exam of the computer science course of the University of Oxford / 2023 / Combinatorial optimisation / 2 / c / v / Solution

Choose a maximal matroid independent set $B\subseteq Y$. Its cardinality is $r(Y)=r(Z)$, so it is also maximal in $Z$. Apply (iv) first inside $Y$ and then inside $Z$:This proves that nested subsets of equal rank of a matroid have the same closure of a matroid.