--- a/hws/hw03.tex	Thu Oct 05 14:36:54 2023 +0100
+++ b/hws/hw03.tex	Fri Oct 13 15:07:37 2023 +0100
@@ -69,7 +69,11 @@
 
   What is the language recognised by this automaton?
 
-  
+  \solution{
+    All strings consisting of 0 or more a's then 1 or more b's,
+    which is equivalent to the language of the regular
+    expression $a^* \cdot b \cdot b^*$.  
+  }
 
 \item Give a non-deterministic finite automaton that can
   recognise the language $L(a\cdot (a + b)^* \cdot c)$.
@@ -109,6 +113,8 @@
         
       }
 
+     
+
 \item Given the following deterministic finite automaton over
       the alphabet $\{a, b\}$, find an automaton that
       recognises the complement language. (Hint: Recall that
@@ -178,6 +184,12 @@
     \end{tikzpicture}
   \end{center}
 
+   \solution{
+        The DFA has three states Q0,Q1,Q2 with Q0 starting state and Q2 accepting.
+        The transitions are (Q0,a)-> Q1 (Q0,b)->Q0 (Q1,a)->Q2 (Q1,b)->Q0
+        (Q2,a)->Q2 (Q2,b)->Q0.
+        }
+  
 \item %%\textbf{(Deleted for 2017, 2018, 2019)}
   Given the following deterministic finite automaton over the
   alphabet $\{0, 1\}$, find the corresponding minimal automaton. In
@@ -232,6 +244,10 @@
   Arden's lemma which states that an equation of the form $q = q\cdot r + s$
   has the unique solution $q = s \cdot r^*$.)
 
+  \solution{
+    $(b + ab + aa(a^*)b)^* \cdot (1 + a)$
+    }
+
 \item If a non-deterministic finite automaton (NFA) has
   $n$ states. How many states does a deterministic 
   automaton (DFA) that can recognise the same language