For (X, Sp) defined in each of the following exercises, describe all independent subsets of X (as defined in D4). (a) Exc. 2; (b) Exc. 3; (c) Exc. 4; (d) Exc. 5; (e) Exc. 6; (f) Exc. 7.

Exercise 2

Let X be any set, and define Sp(S) = ∅ for all S ⊆ X. Which axioms for an independence structure hold?

Exercise 3

Let X be any set, and define Sp(S) = S for all S ⊆ X. Which axioms for an independence structure hold?

Exercise 4

Let X be any set, and define Sp(S) = X for all S ⊆ X. Which axioms for an independence structure hold?

Exercise 5

Uniform Independence Structures. (a) Let X be any set, and fix k ∈ ℕ⁺. For S ⊆ X, define Sp(S) = S if |S|

Exercise 6

Independence Structure for a Set Partition. Suppose I is an index set, {T_i : i ∈ I} are given pairwise disjoint nonempty sets, and X = S _i∈I T_i . Given S ⊆ X, define Sp(S) to be the union of all T_j ’s such that S ∩ T_j ≠ ∅. Prove that axioms A0 through A4 hold.

Exercise 7

Let X = ℝ, and for each S ⊆ X, let Sp(S) =  the closure of S in the real line. (By definition, given z ∈ ℝ and S ⊆ ℝ, z ∈  iff for all ε > 0, the open interval (z − ε, z + ε) intersects S.) Which axioms for an independence structure hold?