Testing notation

math/proof/notes.page

@@ -159,7 +159,7 @@ $2^n$ total leaves of the tree.
 
 **Exercises 1.3**
 
-1. Subsets of \{1,2,3,4\}: $\emptyset, \{1\}, \{2\}, \{3\}, \{4\},
+1. Subsets of $\{1,2,3,4\}$: $\emptyset, \{1\}, \{2\}, \{3\}, \{4\},
    \{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}, \{3,4\}, \{1,2,3\},
    \{1,2,4\}, \{1,3,4\}, \{2,3,4\}, \{1,2,3,4\}$
 
@@ -176,17 +176,24 @@ $A$. I.e. $\mathscr{P}(A) = \{X:X\subseteq A\}$.
 * * * *
 
 **Definition 1.5**
+
 $A$ and $B$ are sets.
-- The *union* of $A$ and $B$ is the set $A\cup B = \{x : x \in A or x
+
-  \in B\}$
+- The *union* of $A$ and $B$ is the set $A\cup B = \{x : x \in A
+  \text{ or } x \in B\}$
 - The *intersection* of $A$ and $B$ is the set $A\cap B = \{x : x \in
-  A and x \in B\}$
+  A \text{ and } x \in B\}$
 - The *difference* of $A$ and $B$ is the set $A - B = \{x : x \in A
-  and x \notin B\}$
+  \text{ and } x \notin B\}$
 
 The operations $\cup$ and $\cap$ obey the commutative law for sets,
 but $-$ does not.
 
+- If an expression involving sets uses only $\cap$ or $\cup$,
+  parentheses are optional.
+
+- If it uses both $\cap$ and $\cup$, parentheses are required!
+
 * * * *
 
 We usually discuss sets in some context. Our sets in that context will
@@ -200,9 +207,26 @@ as the set of points in a circle $C$, the universe would be
 $\Bbb{R}^2$.
 
 **Definition 1.6**
-: $A$ is a set in the universe $U$. The *complement* of $A$ or $\bar A$
+: $A$ is a set in the universe $U$. The *complement* of $A$ or $\overline A$
-is the set $\bar A = U - A$.
+is the set $\overline A = U - A$.
 
 E.g. if $P$ is the set of prime numbers, then $$
-\bar P = \Bbb{N} - P = \{1,4,6,8,9,10,12,\dots\}
+\overline P = \Bbb{N} - P = \{1,4,6,8,9,10,12,\dots\}
-$$ so $\bar P$ is the set of composite numbers and 1.
+$$ so $\overline P$ is the set of composite numbers and 1.
+
+**Definition 1.7**
+
+$A_1, A_2,\dots, A_n$ are sets.
+
+$$
+\align{
+A_1 \cup A_2 \cup A_3 \cup \dots \cup A_n &=
+\left{x : x \in A_i
+  \text{ for at least one set $A_i$, for }
+  1 \leq i \leq n\right}\\
+A_1 \cap A_2 \cap A_3 \cap \dots \cap A-n &=
+\left{x : x \in A_i
+  \text{ for every set $A_i$, for }
+  1 \leq i \leq n \right}\\
+}
+$$