INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS

TLDR

The paper introduces notation where a, b, c, d, x, y, z denote finite sets of non‑negative integers, while other lowercase letters denote non‑negative integers. The authors define a system of finite sets with operations such as union, difference, intersection, and an obliteration operator, introduce the notation [k,I) for specific subsets, and use the function 8(k,l,m) to denote all systems (a0,a1,…,dn) satisfying given inclusion constraints. Reference 2 is cited.

Abstract

2. Notation The letters a, b, c, d, x, y, z denote finite sets of non-negative integers, all other lower-case letters denote non-negative integers. If fc I, then [k, I) denotes the set {k,k+l,k+2,...,l-l} = {*: fc < « < Q. The obliteration operator serves to remove from any system of elements the element above which it is placed. Thus [k,I) = {k,k-{-l,...,fy. The cardinal of o is \a inclusion (in the wide sense), union, difference, and intersection of sets are denoted by o c b, a-\-b, a—b, ab respectively, and a—b = a—ab for all a, b. ), By 8(k,l,m) we denote the set of all systems (ao,av...,dn) such that avc[0,m); \av 1 (v < »),