diff --git a/src/intro/main.tex b/src/intro/main.tex
index bdcfc80..70c3971 100644
--- a/src/intro/main.tex
+++ b/src/intro/main.tex
@@ -39,7 +39,7 @@
 
 Much like in classical error correction, in \ac{qec} information
 is protected by mapping it onto codewords in an expanded
-Hilbert space, thereby introducing redundancy. Take for example, the
+Hilbert space, thereby introducing redundancy. Take, for example, the
 two-qubit code \cite{roffe_quantum_2019}, where we map%
 %
 \begin{align*}
@@ -59,7 +59,7 @@ To determine if an error occurred, we want to know
 whether a state belongs%
 \footnote{
     It is possible for a state to not completely lie in either subspace.
-    In this case, we can interpret the state as being in
+    In this case, we can interpret it as being in
     $\mathcal{C}$ or $\mathcal{F}$ with a certain probability.
 }
 to $\mathcal{C}$ or $\mathcal{F}$.
@@ -69,8 +69,8 @@ described using operators \cite[Section
 1.5]{griffiths_introduction_1995}. Because of the way these operators are
 defined, their eigenvalues correspond to the possible outcomes of
 measuring that observable, and the corresponding
-eigenstates are the states that yield those values as measurements \cite[Section
-3.3]{griffiths_introduction_1995}.
+eigenstates are the determinate states that yield those values as
+measurements \cite[Section 3.3]{griffiths_introduction_1995}.
 In our case, we need an operator with two eigenvalues, and the corresponding
 eigenspaces should be $\mathcal{C}$ and $\mathcal{F}$ respectively.
 For the two-qubit code, $Z_1Z_2$ is such an operator:%
@@ -95,7 +95,7 @@ logical state $\ket{\psi}_\text{L}$, we prepare an ancilla
 qubit with state $\ket{0}_\text{A}$ and we entangle it with
 $\ket{\psi}_\text{L}$ in such a way that determining that instead
 indicates the eigenvalue. More specifically, using a syndrome
-extraction circuit as shown in fig. \ref{fig:syndrome extraction}, we
+extraction circuit as shown in Figure \ref{fig:syndrome extraction}, we
 transform the state of the three-qubit system as%
 
 %
@@ -114,8 +114,8 @@ transform the state of the three-qubit system as%
 If $E \ket{\psi}_\text{L} \in \mathcal{C}$, the second term will
 cancel and we will deterministically measure $\ket{0}_\text{A}$ for
 the ancilla qubit. Similarly, if $E \ket{\psi}_\text{L} \in
-\mathcal{F}$ we will deterministically measure $\ket{1}_\text{A}$.
-In general, the resulting state of the three-qubit system will be a
+\mathcal{F}$, we will deterministically measure $\ket{1}_\text{A}$.
+In general, however, the resulting state of the three-qubit system will be a
 superposition of the two cases.
 Note that the expressions $P_\mathcal{C}$ and $P_\mathcal{F}$ above
 essentially constitute projection operators onto $\mathcal{C}$ and