diff --git a/macros.tex b/macros.tex
index d50997a..13763b8 100644
--- a/macros.tex
+++ b/macros.tex
@@ -3,6 +3,17 @@
 % NOTATION
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+
+%RA-to-Poly Notation
+\newcommand{\rel}{R}
+\newcommand{\reli}{S}
+\newcommand{\relii}{T}
+\newcommand{\query}{Q}
+\newcommand{\join}{\Join}
+\newcommand{\select}{\sigma}
+\newcommand{\project}{\pi}
+\newcommand{\union}{\cup}
+
 %Polynomial Reformulation
 \newcommand{\wbit}{w}
 \newcommand{\expct}{\mathop{\mathbb{E}}}
@@ -324,8 +335,8 @@
 \newcommand{\schemaOf}{\textsc{Sch}}
 \newcommand{\arity}[1]{arity({#1})}
 \newcommand{\db}{D}
-\newcommand{\rel}{R}
-\newcommand{\query}{Q}
+%\newcommand{\rel}{R}
+%\newcommand{\query}{Q}
 \newcommand{\qClass}{\mathcal{C}}
 % \newcommand{\uDom}{\mathcal{U}}
 \newcommand{\tup}{t}
@@ -339,10 +350,7 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % Algebra
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\newcommand{\selection}{\sigma}
-\newcommand{\projection}{\pi}
-\newcommand{\join}{\Join}
-\newcommand{\union}{\cup}
+
 \newcommand{\intersection}{\cap}
 \newcommand{\rename}{\rho}
 \newcommand{\aggregation}[2]{{}_{#1}\gamma_{#2}}
diff --git a/ra-to-poly.tex b/ra-to-poly.tex
index 2ffabfe..d0e9a40 100644
--- a/ra-to-poly.tex
+++ b/ra-to-poly.tex
@@ -1,6 +1,29 @@
 %root: main.tex
 
 \section{Query translation into polynomials}
-This section will involve the set of queries (RA+) that we are interested in, the probabilistic/incomplete models we address, and the outer aggregate functions we perform over the output \textit{annotation}.
+\AH{This section will involve the set of queries (RA+) that we are interested in, the probabilistic/incomplete models we address, and the outer aggregate functions we perform over the output \textit{annotation}
+1) RA notation
+2) DB (TIDB) notation
+3) How queries translate into polynomials
+}
+
+Given tables $\rel, \reli$, an arbitrary query $\query(\rel)$ over the positive relational operators (SPJU), abusing notation slightly denote the query polynomial as $\poly(X_1,\ldots, X_\numTup)$.  The annotation for arbitrary tuple $\tup$ can be viewed as an element of the image of $\rel$, where relation $\rel$ can be thought of as a function with preimage of all tuples in $\rel$, such that $\rel(\tup) = \poly(X_1,\ldots, X_\numTup)$.  Further, it is known that the algebraic semiring structure aptly models the translation and computation of query operations into tuple annotation, aka polynomials.  
+To make things more concrete, consider the $\{\mathbb{N}, \times, +, 1, 0\}$ bag semiring.  Here the set in which the tuple annotations (computed polynomials) exist is the natural numbers.  Query operations are translated into one of the two semiring operators, with $\project$ and $\union$ of agreeing tuples being the equivalent of the '+' opertator in polynomial $\poly$, $\join$ translating into the $\times$ operator, and finally, $\select$ is better modeled as a function that returns either $\rel(\tup)$ or $0$ based on some predicate.
+
+Consider the translation of relational operators to polynomial operators in greater detail.
+
+\begin{align*}
+&\project_A(\rel)(\tup) = &&\sum_{\tup' s.t. \tup[A] = \tup'} \rel(\tup')\\
+& (\rel_1 \union \rel_2)(\tup) = &&\rel_1(\tup) + \rel_2(\tup)\\
+&(\rel_1 \join_\theta \rel_2)(\tup) = &&\begin{cases}
+						\rel_1(\tup_1) \times \rel_2(\tup_2)	&\text{if }\theta(\tup_1, \tup_2)\\
+						0						&\text{otherwise}
+					 \end{cases} \\
+&\select_\theta(\rel) = &&\begin{cases}
+					\rel(\tup)	&\text{if }\theta(\tup) = 1\\
+					0		&\text{otherwise}.
+				\end{cases}
+\end{align*}
+
 
 \AR{This section needs to be written up {\bf before} you go forwad in the next section. In particular, this section should setup definitions/notation (e.g. how you to denote a general query on TIDB) that will be used later on so better get thet setup first.}