diff --git a/source/logic/sec-tautology.ptx b/source/logic/sec-tautology.ptx
index 5d29b50..f968e7b 100644
--- a/source/logic/sec-tautology.ptx
+++ b/source/logic/sec-tautology.ptx
@@ -2,66 +2,68 @@
 
 <section xml:id="sec-tautology" xmlns:xi="http://www.w3.org/2001/XInclude">
     <title>Analyzing Logical Equivalences</title>
-<subsection>
-    <title>Equivalent Statements</title>
-    <p>
-        By comparing the truth tables, we can ascertain if two logical statements are equivalent, meaning they have identical truth values for all possible inputs.
-    </p>
-    <aside>
-        <title>Notes</title>
+    <subsection>
+        <title>Equivalent Statements</title>
+        <p>
+            When working with Sage symbolic logic, the <c>==</c> operator compares semantic equivalence.
+        </p>
+        <sage>
+            <input>
+                h = propcalc.formula("x | ~y")
+                s = propcalc.formula("x &amp; y | x &amp; ~y | ~x &amp; ~y")
+                h == s
+            </input>
+        </sage>
+        <p>
+            Do not attempt to compare equivalence of truth tables.
+        </p>
+        <sage>
+            <input>
+                # Warning:
+                # Even though these truth tables look identical,
+                # the comparison will return False.
+                h.truthtable() == s.truthtable()
+            </input>
+        </sage>
         <p>
-            This approach is invaluable in simplifying logical expressions in computer algorithms and understanding proofs in mathematical logic.
+            However, we can compare equivalence of truth table <c>lists</c>.
         </p>
-    </aside>
-    <sage>
-        <input>
-            E1 = propcalc.formula('(p -&gt; q) &amp; (q -&gt; r)')
-            E2 = propcalc.formula('p -&gt; r')
-            T1 = E1.truthtable()
-            T2 = E2.truthtable()
-            T1 == T2
-        </input>
-        <output></output>
-    </sage>
-</subsection>
+        <sage>
+            <input>
+                h_list = h.truthtable().get_table_list()
+                s_list = s.truthtable().get_table_list()
+                h_list == s_list
+            </input>
+        </sage>
+    </subsection>
     <subsection>
-    <title>Tautologies</title><idx><h>tautology</h></idx>
-    <p>
-        A tautology is a logical statement that is always true.  The <c>is_tautology()</c> function checks whether a given logical expression is a tautology.
-    </p>
-    <aside>
-        <title>Notes</title>
+        <title>Tautologies</title><idx><h>tautology</h></idx>
         <p>
-            Understanding tautologies is essential in computer science, particularly in optimizing algorithms and validating logical propositions in software verification. It's also fundamental in mathematical proof strategies, such as proof by contradiction.
+            A tautology is a logical statement that is always true.  The <c>is_tautology()</c> function checks whether a given logical expression is a tautology.
         </p>
-    </aside>
-    <sage>
-        <input>
-            a = propcalc.formula('p | ~p')
-            a.is_tautology()
-        </input>
-        <output></output>
-    </sage>
+        <aside>
+            <title>Notes</title>
+            <p>
+                Understanding tautologies is fundamental in mathematical proof strategies, such as proof by contradiction.
+            </p>
+        </aside>
+        <sage>
+            <input>
+                a = propcalc.formula('p | ~p')
+                a.is_tautology()
+            </input>
+        </sage>
     </subsection>
-
     <subsection>
-    <title>Contradictions</title><idx><h>contradiction</h></idx>
-    <p>
-        In contrast to tautologies, contradictions are statements that are always false.
-    </p>
-    <aside>
-        <title>Notes</title>
+        <title>Contradictions</title><idx><h>contradiction</h></idx>
         <p>
-            Analyzing the relationship between tautologies and contradictions can aid in the development of critical thinking skills, which are essential for debugging in programming, constructing mathematical proofs, and designing logical circuits.
+            In contrast to tautologies, contradictions are statements that are always false.
         </p>
-    </aside>
-    <sage>
-        <input>
-            A = propcalc.formula('p &amp; ~p')
-            A.is_contradiction()  
-        </input>
-        <output></output>
-    </sage>
+        <sage>
+            <input>
+                A = propcalc.formula('p &amp; ~p')
+                A.is_contradiction()  
+            </input>
+        </sage>
     </subsection>
-
 </section>