wiki/math/proof/notes.page

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title: Mathematical Proof Study Notes
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# 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:
$$ A_1 \times A_2 \times \dots \times A_n = \{
 (x_1,x_2,\dots,x_n) : x_i \in A_i, i \in \Bbb{N} \}
$$

We can also take *Cartesian powers* of sets. For a set $A$ and
positive integer $n$, $A^n$ is the Cartesian product of $A$ with
itself $n$ times: $$
A^n = A \times A \times \dots \times A = \{(x_1,x_2,\dots,x_n):x_i \in
A,i \in \{1,\dots,n\}\} $$

* * * *

**Exercises 1.2**

1. $A = \{1,2,3,4\}, B = \{a,c\}$
    a. $A \times B =
    \{(1,a),(2,a),(3,a),(4,a),(1,c),(2,c),(3,c),(4,c)\}$
    b. $B \times A =
    \{(a,1),(c,1),(a,2),(c,2),(a,3),(c,3),(a,4),(c,4)\}$
    d. $B \times B = \{(a,a),(a,c),(c,a),(c,c)\}$
    e. $\emptyset \times B = \emptyset$
    f. $(A \times B) \times B = \{((1,a),a),((2,a),a),
       ((3,a),a),((4,a),a),((1,c),a),((2,c),a),((3,c),a),
       ((4,c),a),((1,a),a),((2,a),c),((3,a),c),((4,a),c),
       ((1,c),c),((2,c),c),((3,c),c),((4,c),c)\}$

* * * *

**Definition 1.3**
: $A$ and $B$ are sets. If every element of $A$ is also an element of
$B$, then $A$ is a *subset* of $B$ and we write $A \subseteq B$. If
this is not the case, we write $A \not\subseteq B$, which means there is
at least one element of $A$ that is not in $B$.

**Fact 1.2**
: It follows from **1.3** that for any set $B$, $\emptyset \subseteq
B$. I.e., the empty set is a subset of every set.

**Fact 1.3**
: If a finite set has $n$ elements, it has $2^n$ subsets.

This can be shown by drawing a decision tree starting with the empty
set, with each fork representing a choice of whether to insert the
next element of the set in question. Since there are two possibilities
at each fork and $n$ elements to consider for insertion, that gives
$2^n$ total leaves of the tree.

* * * *

**Exercises 1.3**

1. Subsets of $\{1,2,3,4\}$: $\emptyset, \{1\}, \{2\}, \{3\}, \{4\},
   \{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}, \{3,4\}, \{1,2,3\},
   \{1,2,4\}, \{1,3,4\}, \{2,3,4\}, \{1,2,3,4\}$

* * * *

**Definition 1.4**
: $A$ is a set. The *power set* of $A$ is another set,
$\mathscr{P}(A)$, defined to be the set of all subsets of
$A$. I.e. $\mathscr{P}(A) = \{X:X\subseteq A\}$.

**Fact 1.4**
: If $A$ is a finite set, $|\mathscr{P}(A)| = 2^{|A|}$.

* * * *

**Definition 1.5**

$A$ and $B$ are sets.

- The *union* of $A$ and $B$ is the set $A\cup B = \{x : x \in A
  \text{ or } x \in B\}$
- The *intersection* of $A$ and $B$ is the set $A\cap B = \{x : x \in
  A \text{ and } x \in B\}$
- The *difference* of $A$ and $B$ is the set $A - B = \{x : x \in A
  \text{ and } x \notin B\}$

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
naturally be subsets of some other set, which we call the *universal
set* or just *universe*. If we don't know specifically which set it
is, we call it $U$.

For example, when discussing the set of prime numbers $P$, the
*universal set* is $\Bbb{N}$. When we discuss geometric figures such
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 $\overline A$
is the set $\overline A = U - A$.

E.g. if $P$ is the set of prime numbers, then $$
\overline P = \Bbb{N} - P = \{1,4,6,8,9,10,12,\dots\}
$$ so $\overline P$ is the set of composite numbers and 1.

**Definition 1.7**

$A_1, A_2,\dots, A_n$ are sets. Then

$$\begin{align}
A_1 \cup A_2 \cup A_3 \cup \dots \cup A_n &=
\left\{x : x \in A_i
  \text{ for at least one set $A_i$, for }
  1 \leq i \leq n\right\}\\
A_1 \cap A_2 \cap A_3 \cap \dots \cap A_n &=
\left\{x : x \in A_i
  \text{ for every set $A_i$, for }
  1 \leq i \leq n \right\}\\
\end{align}$$

Given $A_1, A_2, \dots, A_n$ we define

$$\begin{align}
\bigcup{i=1}{n} A_i &= A_1 \cup A_2 \cup \dots \cup A_n\\
\bigcap{i=1}{n} A_i &= A_1 \cap A_2 \cap \dots \cap A_n\\
\end{align}$$

$$
A_1 = \{-1,0,1}, A_2 = \{-2,0,2}, \dots, A_i = \{-i,0,i}, \dots\\
\bigcup{i=1}{\inf} A_i = \Bbb{Z} \text{ and }
\bigcap{i=1}{\inf} A_i = \{0\}\\
$$