403/503 – Homework 1

February 3, 2010

A collection {\mathcal I} of subsets of {\mathbb N} is called an independent family iff for any pairwise distinct A_1,\dots,A_k,B_1,\dots,B_n in {\mathcal I}, we have that the set A_1\cap\dots\cap A_k\cap({\mathbb N}\setminus B_1)\cap\dots\cap({\mathbb N}\setminus B_n) is infinite.

  1. Show that there is an independent family of the same size as the reals. (This means that there is an independent family {\mathcal I} for which there is a bijection between {\mathcal I} and {\mathbb R}. If it is easier to think of it this way, you can use that {\mathbb R} has the same size as the set 2^{\mathbb N} of infinite sequences of zeros and ones.)
  2. Given a set A\subseteq{\mathbb N}, recall that its characteristic function is the map \chi_A:{\mathbb N}\to\{0,1\} such that \chi_A(n)=1 if n\in A, and \chi_A(n)=0 if n\notin A. We can think of each \chi_A as a vector in the space {\mathbb R}^{\mathbb N}. Suppose now that {\mathcal I} is an independent family, and show that \{\chi_A\mid A\in{\mathcal I}\} is a linearly independent set. (Thus any basis for {\mathbb R}^{\mathbb N} over {\mathbb R} must have the same size as the reals.)
  3. Consider {\mathbb R} as a vector space over {\mathbb Q}, and show that any basis must have the same size as the reals.