More notes

math/proof/notes.page

@@ -37,6 +37,8 @@ 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**
 
 $\Bbb{N}$
@@ -46,8 +48,8 @@ $\Bbb{Z}$
 : *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}
+: *rational numbers*, $\Bbb{Q} = \{x : x = \frac mn; m,n \in \Bbb{Z};
-and n \neq 0\}$.
+n \neq 0\}$.
 
 $\Bbb{R}$
 : *real numbers*, the set of all real numbers on the number line.
@@ -66,3 +68,46 @@ the set, which is the number of elements it has. E.g. $|A| = 4$.
  $$
 This can be read as "E is the set of all things of form $2n$, such
  that $n$ is an element of $\Bbb{Z}$."
+
+*intervals*
+: For two numbers $a,b \in \Bbb{R}$ with $a < b$, we can form various
+intervals on the number line by listing them as a bracketed pair. A
+parenthesis indicates that side of the interval is *open*, while a
+square bracket indicates that side of the interval is *closed*. A
+closed interval *includes* the element of the pair on the closed side,
+while an open interval does not. Infinite intervals are denoted by
+including $\inf$ as one member of the pair on the open side.
+
+* * * *
+
+**Exercises 1.1**
+
+1. $\{4x-1 : x \in \Bbb{Z}\}$ is
+   $\{\dots, -21, -16, -11, -6, -1, 4, 9, 14, 19,\dots \}$
+
+* * * *
+
+With two sets $A$ and $B$, one can "multiply" them to form the set $A
+\times B$ which is called the *Cartesian product*.
+
+**Definition 1.1**
+: An *ordered pair* is a list $(x,y)$ of two things $x$ and $y$,
+enclosed in parentheses and separated by a comma. They are
+distinguished by order; e.g. $(3,4) \neq (4,3)$.
+
+**Definition 1.2**
+: The *Cartesian product* of two sets $A$ and $B$ is another set, $A
+\times B$, defined as $A \times B = \{(a,b):a \in A, b \in B\}$.
+
+E.g. $\{0,1\} \times \{2,1\} = \{(0,2),(0,1),(1,2),(1,1)\}$
+
+**Fact 1.1**
+: If $A$ and $B$ are finite sets, then $|A \times B| = |A| \cdot |B|$.
+
+The set $\Bbb{R} \times \Bbb{R}$ can represent the points on the
+Cartesian plane.
+
+The idea extends to a 3-list, or *ordered triple*. In general:
+$$ A_1 \times A_2 \times \dots \times A_n = \{
+ (x_1,x_2,\dots,x_n) : x_i \in A_i, i \in \Bbb{N} \}
+$$