114 lines
3.1 KiB
Plaintext
114 lines
3.1 KiB
Plaintext
---
|
|
format: markdown
|
|
toc: yes
|
|
title: Mathematical Proof Study Notes
|
|
...
|
|
|
|
# Texts
|
|
|
|
- *Book of Proof*, Richard Hammack
|
|
|
|
# Reading Notes
|
|
|
|
## *Hammack*, 25 Jan 2014
|
|
|
|
All of Mathematics can be described with *sets*.
|
|
|
|
*set*
|
|
: A collection of things. The things in the set are called *elements*.
|
|
|
|
An example of a set: $\{2,4,6,8\}$
|
|
|
|
The set of all integers:
|
|
$$ \{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\} $$
|
|
|
|
The dots mean the expressed pattern continues.
|
|
|
|
Sets of infinitely many members are *infinite*, otherwise they are
|
|
*finite*.
|
|
|
|
Sets are *equal* if they have exactly the same elements.
|
|
|
|
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*, $\Bbb{Q} = \{x : x = \frac mn; m,n \in \Bbb{Z};
|
|
n \neq 0\}$.
|
|
|
|
$\Bbb{R}$
|
|
: *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 $\Bbb{Z}$."
|
|
|
|
*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:
|
|
$$ A_1 \times A_2 \times \dots \times A_n = \{
|
|
(x_1,x_2,\dots,x_n) : x_i \in A_i, i \in \Bbb{N} \}
|
|
$$
|