\documentclass[12pt]{article}
\usepackage{mymacros}

%1.2: 4,5
%1.3: 1,3

\begin{document}
\title{1.2-1.3}
\author{Andrew Lounsbury}
\date{January 27, 2020}
\maketitle

\section{1.2, p.7}

\begin{enumerate}
	\item[4.]
	variables: $x,y,t$ \\
	constants: $\{\ol{0}, \ol{1},\dots,\ol{16}\}$ \\
	$R_1:$ (team) $x R_1$ (team) $y$ if $x$ and $y$ have played at home. \\
	$R_2:~x R_2 y$ if $x$ and $y$ have played away. \\
	$R_3:~x R_3 t$ if $x$ played on $t$. \\
	%$R_3:~x R_3 y$ if $x$ and $y$ have played once at home and once on the road. 
	\[
		\neg(\exists x)(\exists y)(\exists t)[x R_3 t \land y R_3 t]
	\]

	\item[5.]
	constants: $\{\ol{0}, \ol{1}, \ol{2}, \dots, \ol{10}, \ol{11}, \ol{12},\dots; \ol{0.1}, \ol{0.12}, \ol{0.13}, \dots, \ol{0.2}, \ol{0.21}, \ol{0.22}, \dots\}$

	variables: $\{t_1,t_2,\dots,t_n,s_1,s_2,\dots,s_{3n},a_1,a_2,\dots,a_{3n}\}$

	$\{+,\cdot\}$; arity 2 \\
	$\{\sin,\cos,\tan,\csc,\sec,\cot\}$; arity 1 \\
	$\{\sim,\cong,=\}$

\end{enumerate}

\section{1.3, p.12}

\begin{enumerate}
	\item[1.]
	$\mathcal{L} = \{\diamondsuit, \heartsuit; Y^1;\flat^2,\#^3\}$ \\
	terms: $\diamondsuit, \heartsuit, \flat\diamondsuit\heartsuit, \#\diamondsuit\heartsuit(\flat\diamondsuit\heartsuit)$ \\
	non-terms: $Y, Y\diamondsuit, Y\heartsuit$ \\
	formulas: $=\diamondsuit\heartsuit, Y\diamondsuit, \neq\diamondsuit\heartsuit, (=\diamondsuit\heartsuit \lor Y\diamondsuit\heartsuit)$ \\
	non-formulas: $\flat\diamondsuit\heartsuit, \#\diamondsuit\heartsuit(\flat\diamondsuit\heartsuit), (\flat\diamondsuit\heartsuit \lor Y\diamondsuit\heartsuit)$

	\item[3.]
	$\mathcal{L}_{NT}$ is $\{0,S,+,\cdot,E,<\}$
	\begin{enumerate}
	\item
	
	\item
	\begin{align*}
	(\forall x)(\exists y)(\exists z)&(\exists w)(\exists v)(\exists k)(\exists \ell)(\exists m)(\exists n) \\
	&[x = y + z + w + v \sland y = k^2 \sland z = \ell^2 \sland w = m^2 \sland v = n^2]
	\end{align*}
	
	\item
	
	\item
	
	\item
	
	\end{enumerate}

\end{enumerate}

\end{document}