 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2023/combinatorial-optimisation/2/c/v/solution

B \subseteq Y

r (Y) = r (Z)

, so it is also maximal in

Z

. Apply (iv) first inside

Y

and then inside

Z

σ (Y) = σ (B) = σ (Z) .

This proves that nested subsets of equal rank of a matroid have the same closure of a matroid.

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