Tags on Subsets

Let X be a finite set of v elements. Let X = {x1, x2,…,xv}. We will assume that X is totally ordered, x1 < x2 < … < xv. Let Y ⊆ X, Y = {y1, y2,…, y1}. Unless stated otherwise, we will assume that Y is a chain, written in increasing order, i.e., y1 < y2 < … < l. We will denote by P(X), the set all subsets of X and Pk(X), the set of all k-subsets of X, 0 ≤ k ≤ v. We will denote by Vk(X), the set of all rational valued functions f : Pk(X) → Q. Clearly Vk(X) is a vector space over Q, of dimension (v k). The set of Mk(X) ⊆ Vk(X) of all integral valued functions, is clearly a module of rank (v k) over the ring of integers Z.