More notes

math/proof/notes.page

@@ -34,3 +34,35 @@ E.g. $\{2,4,5,8\} = \{4,2,8,6\}$ but $\{2,4,6,8\} \neq \{2,4,6,7\}$.
 Uppercase letters often denote sets, e.g. $A = \{1,2,3,4\}$.
 
 To express membership, we use $\in$, as in $2 \in A$.
+
+To express non-membership, we use $\notin$, as in $5,6 \notin A$.
+
+**Special Sets**
+
+$\Bbb{N}$
+: *natural numbers*, the positive whole numbers $\{1,2,3,4,5,\dots\}$
+
+$\Bbb{Z}$
+: *integers*, $\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4,\dots\}$
+
+$\Bbb{Q}$
+: *rational numbers*, $\Bbb{Q} = \{x : x = \frac mn, m,n \in \Bbb{Z}
+and n \neq 0\}$.
+
+$\Bbb{R}$
+: *real numbers*, the set of all real numbers on the number line.
+
+$\emptyset$
+: *empty set*, the unique set with no members, $\{\}$
+
+* * * *
+
+For finite sets $X$, $|X|$ represents the *cardinality* or *size* of
+the set, which is the number of elements it has. E.g. $|A| = 4$.
+
+*set-builder notation* describes sets that are too big or complex to
+ be listed out. E.g. the infinite set of even integers: $$
+ E = \{2n : n \in \Bbb{Z}\}
+ $$
+This can be read as "E is the set of all things of form $2n$, such
+ that $n$ is an element of $\Bbb{Z}$."