## 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.