In my Sage/CoCalc article (Sage has recently merged with CoCalc), I am unable to create an Array without a red issue bar popping up. Morover, I am unable to cross out element in my array. The issue(s) is PackageArrayError; illegal pream-token (1): PackageArrayError; illegal pream-token (2): ...... and so on.
The whole Document is below.
\documentclass{article}
% set font encoding for PDFLaTeX or XeLaTeX
\usepackage{ifxetex}
\ifxetex
\usepackage{fontspec}
\else
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\fi
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{ragged2e}
\usepackage{graphicx}
\usepackage{breqn}
\usepackage{fixltx2e}
\usepackage{hyperref}
\usepackage{titlesec}
\usepackage{amsopn}
\usepackage{array}
\newcommand{\dd}[1]{\mathrm{d}#1}
% used in maketitle
\title{Regarding Your Version of Natural Density to Rationals}
\author{Aiyappan Nair}
\begin{document}
\maketitle
Your version of asymptotic density to rationals differs from my own version. It is not a simplification but rather a change in definition. And although it is much more simple there are flaws that I would like to mention.
\subsection{Counter-Example}
As mentioned earlier, if sets $T_1,T_2,T_p$ are subsets of $\mathbb{Q}$, a density $D$ should have the following qualities.
(1) If $T_1=T_2$ then $D(T_1)=D(T_2)$
(2) If $T_1\subset{T_2}$ then $D(T_1)\le D(T_2)$
In your new version of the density
$$D=\lim_{r\to\infty}\frac{\left|T_1\cap V(r)\right|}{|V(r)|}$$
Where
$$V(r)=\left\{\left.\frac{m}{n}\right|1<m<r,1<n<r,\gcd(m,n)=1\right\}$$
and $m,n,r\in\mathbb{Z}$
If we set $T_1=\left\{\left.\frac{m}{2n+1}\right|m,n\in\mathbb{Z}\right\}$ and $T_2=\left\{\left.\frac{m}{4n+2}\right|m,n\in\mathbb{Z}\right\}$, I found that $D(T_1)>D(T_2)$ even though $T_1\subset T_2$. Which breaks requirment (2).
\subsection{Proof of Counter Example}
Imagine $V(r)$ as an array of values, restricted by $r$. The numerator is from listed as columns each labelled from $[1,r]$ and the denominator is listed as row labelled from $[1,r]$.
\begin{array}{|l|col1 col2 col3 col4 col5 col6 col7 col8}
& 1 & 2 & 3 & 4 & 5 & .. & r\\
\hline
1 & 1/1 & 2/1 & 3/1 & 4/1 & 5/1 & ..& r/1\\
2 & 1/2 & \st{2/2} & 3/2 & 4/2 & 5/2 & ..& r/2\\
3 & 1/3 & 2/3 & 3/3 & 4/3 & 5/3 & ..& r/3\\
4 & 1/4 & 2/4 & 3/4 & 4/4 & 5/4 & ..& r/4\\
5 & 1/5 & 2/5 & 3/5 & 4/5 & 5/5 & ..& r/5\\
..& .. & .. & .. & .. & .. & ..& .. \\
r & 1/r & 2/r & 3/r & 4/r & 5/r & .. & r/r\\
\end{array}
\end{document}
I have never had this problem with SageMath before it merged with CoCalc. Please help.
