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for all , .

An alternative definition of the dual matroid is that its basis sets are the complements of the basis sets of . The basis exchange axiom, used to define matroids from their bases, is self-complementary, so the dual of a matroid is necessarily a matroid.

The flats of are complementary to the cyclic sets (unions of circuits) of , and vice versa.

If is the rank function of a matroid

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