[TI] Compact: Add some notes

semester3/ti-compact/parts/01_words-alphabets.tex

@@ -31,10 +31,16 @@ whereas $L_\lambda$ is the language with just the empty word in it.
 
 \bi{Cleen Star}: $L^* = \bigcup_{i \in \N} L^i$ and $L^+ = L \cdot L^*$
 
+Of note is that there are irregular languages whose Cleen Star is regular, most notably,
+the language $L = \{ w \in \{ 0 \}^* \divides |w| \text{ is prime} \}$'s Cleen Star is regular,
+due to the fact that the prime factorization is regular
+
 
 \inlinelemma $L_1L_2 \cup L_1 L_2 = L_1(L_2 \cup L_3)$
 \inlinelemma $L_1(K_2 \cap L_3) \subseteq L_1 L_2 \cap L_1 L_3$
 
+For multiple choice questions, really think of how the sets would look to determine if they fulfill a requirement.
+
 
 \stepcounter{subsection}
 \subsection{Kolmogorov-Complexity}

semester3/ti-compact/parts/02_finite-automata.tex

@@ -54,7 +54,7 @@ For all of them start by assuming that $L$ is regular.
 \setLabelNumber{lemma}{3}
 \begin{lemma}[]{Regular words}
     Let $A$ be a FA over $\Sigma$ and let $x \neq y \in \Sigma^*$, such that $\hdelta_A (q_0, x) = \hdelta(q_0, y)$.
-    Then for each $z \in \Sigma^*$ there exists a $r \in Q$, such that $xz, yz \in \class[r]$, and we thus have
+    Then for each $z \in \Sigma^*$ there exists an $r \in Q$, such that $xz, yz \in \class[r]$, and we thus have
     \rmvspace
     \begin{align*}
         xz \in L(A) \Longleftrightarrow yz \in L(A)