Show that if (S, ) is a matroid, then (S, ) is a matroid, where A S A contains some maximal A That is, the maximal independent sets of (S, ) are just the complements of the maximal independent sets of (S, ) ...

We need to show three things to prove that S is a matroid 1 S is finite We are given that 2 is hereditary Suppose that B e and A B Since B e there is ...

Show that if (S, ) is a matroid, then (S, ) is a matroid, where = {A:

Question:

Show that if (S, ℓ) is a matroid, then (S, ℓ′) is a matroid, where ℓ′ = {A′: S - A′ contains some maximal A ⁪ ¬ℓ}.
That is, the maximal independent sets of (S, ℓ′) are just the complements of the maximal independent sets of (S, ℓ).