auto-update(nvim): 2025-01-07 17:58:53

documents/by-course/math-8/course-notes/main.typ

@@ -95,7 +95,7 @@ as *propositional forms*.
 DeMorgan's Laws tell us how to distribute logical connectives across
 parentheses.
 
-#theorem[DeMorgan's Laws][
+#fact[DeMorgan's Laws][
   + $not (P or Q) = not P and not Q$
   + $not (P and Q) = not P or not Q$
 ]
@@ -104,4 +104,15 @@ parentheses.
   Trivially, by completing a truth table.
 ]
 
+Also, propositional forms obey commutative, associative, distributive laws,
+which can be trivially obtained from symbolic manipulations and will not be
+restated. Together with the double negation law and the _law of the excluded
+middle_, these comprise the axioms of a system of propositional logic.
 
+#fact[
+  We abbreviate propositional forms by eliding parentheses, according to the rules:
+
+  + $not$ is applied to the smallest proposition following it.
+  + $and$ connects the smallest propositions surrounding it.
+  + $or$ connects the smallest propositions surrounding it.
+]

documents/by-course/pstat-120a/course-notes/main.typ

@@ -91,11 +91,11 @@ the previous ones.
 Check your intuition that this makes sense. Why do you think $RR^n$ was chosen
 as the notation for $n$ dimensional spaces in $RR$?
 
-#remark[Disjoint sets][
+#definition[Disjoint sets][
   If $A sect B$ = $emptyset$, then we say that $A$ and $B$ are *disjoint*.
 ]
 
-#fact[Properties of set operations][
+#fact[
   For any sets $A$ and $B$, we have DeMorgan's Laws:
   + $(A union B)' = A' sect B'$
   + $(A sect B)' = A' union B'$
@@ -148,3 +148,12 @@ When a set is uncountably infinite, its cardinality is greater than $aleph_0$.
   every set with cardinality $aleph_0$ has a bijection to $ZZ$. More generally,
   any sets with the same cardinality have a bijection between them.
 ]
+
+This gives us the following equivalent statement:
+
+#fact[
+  Two sets have the same cardinality if and only if there exists a bijective
+  function between them. In symbols,
+
+  $ N(A) = N(B) <==> exists F : A <-> B $
+]