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Levi Pearson 2014-01-25 14:23:04 -07:00
parent a2cc860dc9
commit f3de4b22d0
1 changed files with 47 additions and 2 deletions

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@ -37,6 +37,8 @@ To express membership, we use $\in$, as in $2 \in A$.
To express non-membership, we use $\notin$, as in $5,6 \notin A$. To express non-membership, we use $\notin$, as in $5,6 \notin A$.
* * * *
**Special Sets** **Special Sets**
$\Bbb{N}$ $\Bbb{N}$
@ -46,8 +48,8 @@ $\Bbb{Z}$
: *integers*, $\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4,\dots\}$ : *integers*, $\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4,\dots\}$
$\Bbb{Q}$ $\Bbb{Q}$
: *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\}$.
$\Bbb{R}$ $\Bbb{R}$
: *real numbers*, the set of all real numbers on the number line. : *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 This can be read as "E is the set of all things of form $2n$, such
that $n$ is an element of $\Bbb{Z}$." 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:
$$ A_1 \times A_2 \times \dots \times A_n = \{
(x_1,x_2,\dots,x_n) : x_i \in A_i, i \in \Bbb{N} \}
$$