Definition 2.30 defines a partition of a set S as a set {A1, A2, . . . , Ak} of sets such that (i) A1, A2, . . . , Ak are all nonempty; (ii) A1 ∪ A2 ∪ ∪ Ak = S; and (iii) for any distinct i, j ∈ {1, ....

Definition 2.30 defines a partition of a set S as a set {A₁, A₂, . . . , Ak} of sets such that (i) A₁, A₂, . . . , Ak are all nonempty; (ii) A₁
∪ A₂
∪ ∪ Ak = S; and (iii) for any distinct i, j ∈ {1, . . . , k}, the sets Ai and Aj are disjoint. Formalize this definition using nested quantifiers and basic set notation.