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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

### Ch 1, Sets

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

$$\begin{align}
\bigcup_{i=1}^{n} A_i &= A_1 \cup A_2 \cup \dots \cup A_n\\
\bigcap_{i=1}^{n} A_i &= A_1 \cap A_2 \cap \dots \cap A_n\\
\end{align}$$

For example, take the following infinite list of sets:

$$
A_1 = \{-1,0,1\}, A_2 = \{-2,0,2\}, \dots, A_i = \{-i,0,i\}, \dots
$$

We can see that:

$$
\bigcup_{i=1}^{\inf} A_i = \Bbb{Z} \text{ and }
\bigcap_{i=1}^{\inf} A_i = \{0\}\\
$$

**Definition 1.8**

If we have a set $A_{\alpha}$ for every $\alpha$ in some index set $I$,
then

$$\begin{align}
\bigcup_{\alpha \in I} A_{\alpha} &= \{ x : x \in A_{\alpha} \text {
for at least one set $A_{\alpha}$ with } \alpha \in I \}\\
\bigcap_{\alpha \in I} A_{\alpha} &= \{ x : x \in A_{\alpha} \text {
for every set $A_{\alpha}$ with } \alpha \in I \}\\
\end{align}$$

* * * *

**Assumptions about Sets that are Number Systems**

In this text, the familiar commutative, associative, and distributive
properties of arithmetic operations on numbers are taken for granted
as axioms that we may use in proofs.

We also accept as fact the natural ordering of elements in $\Bbb{N}$,
$\Bbb{Z}$, $\Bbb{Q}$, and $\Bbb{R}$. This includes the *well-ordering
principle*, i.e. that any non-empty subset of $\Bbb{N} has a smallest
element.

**Fact 1.5 (The Division Algorithm)**
: Given integers $a$ and $b$ with $b > 0$, there exist integers $q$
and $r$ for which $a = qb + r$ and $0 /leq r < b$. This follows easily
from the *well-ordering principle*.

**Russel's Paradox**

Consider the set defined to be all sets that do not contain
themselves as elements.

$$
A = \{X : X \text{ is a set and } X \notin X \}
$$

Now consider the set $X = \{\{\{\{\dots\}\}\}\}$, or $X = \{X\}. It is
the single-element set of the single-element set, etc. nesting
infinitely. Its only element is $X$ itself, so $X \in X$. Since $X$
does not satisfy the prerequisite for inclusion in $A$, which is $X
\notin X$, then $X \notin A$.

But is $A$ an element of $A$?

For a set $X$, the definition of $A$ says that $X \in A$ means the
same thing as $X \notin X$. So, if we substitute $A$ for $X$, then the
definition of $A$ must says that $A \in A$ means the same thing as $A
\notin A$.

If $A \in A$ is true, then it is false; if $A \in A$ is false, then it
is true. Paradox!

Mathematicians eventually settled on a set of axioms called the
*Zermelo-Fraenkel axioms*. It includes the *well-ordering principle*
and also an axiom that states that a non-empty set $X$ is not allowed
to have the property $X \cap x \neq \emptyset$ for all its elements
$x$ that are sets. This prevents defining $X = \{X\}$.

### Ch 2, Logic

Logic is a systematic way of thinking that allows us to deduce new
information from old information and to parse the meaning of
sentences.

Following logic allows one to deduce information correctly, but does
not imply that all correct deductions produce correct information.

Correct deduction from incorrect facts will lead to new facts that are
likely to be incorrect.

In proving theorems, we apply logic to information considered
obviously true or to information already proved to be true; then
anything we deduce with correct logic will also be true (at least so
far as our assumptions were correct).

**Statements**

A statement is a sentence or mathematical expression that is either
definitely true or definitely false.

We often use capital letters ($P, Q, R, S$) to stand for specific
statements.

We may use variables in statements. We use the form $P(x)$ to describe
a statement $P$ that involves variable $x$.

A statement whose truth depends on the value of one or more variables
is an *open sentence*.

**And, Or, Not**

We can combine two logical statements together into a new statement.

One way is with *and*, denoted $\land$. If both statements combined
with *and* are true, the resulting statement is also true. If either
is false, the resulting statement is false.

Another way is with *or*, denoted $\lor$. If either statement is true,
the resulting statement is true. If both are false, then the resulting
statement is also false.

Any statement can have its truth value inverted by applying *not* to
it. If we have a true statement $P$, $~P$ is false.  If we have a
false statement $Q$, $~Q$ is true.

**Conditional Statements**

Given two statements $P$ and $Q$, we can make a new statement, *if*
$P$, *then* $Q$. We write such a *conditional statement* as $P \implies
Q$. In terms of logic, $P \implies Q$ is true *unless* $P$ is true but
$Q$ is not.

Note that if $R: P \implies Q$ and $P$ is false, then $R$ is true.

Alternate phrasings of $P \implies Q$:

- If $P$, then $Q$
- $Q$ if $P$
- $Q$ whenever $P$
- $Q$, provided that $P$
- Whenever $P$, then also $Q$
- $P$ is a sufficient condition for $Q$
- For $Q$, it is sufficient that $P$
- $Q$ is a necessary condition for $P$
- For $P$, it is necessary that $Q$
- $P$ only if $Q$

**Biconditional Statements**

$R: P \implies Q$ is not the same as $S: Q \implies P$. We call $S$ the
*converse* of $R$.

When a statement is conditional upon a condition $P$ and its converse
$Q$, we use the phrase $P$ *if and only if* $Q$. We write it $P \iff
Q$.

Alternate phrasings of $P \iff Q$

- $P$ if and only if $Q$
- $P$ is a necessary and sufficient condition for $Q$
- For $P$ it is necessary and sufficient that $Q$
- If $P$, then $Q$, and conversely.