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Levi Pearson 2014-01-25 17:10:38 -07:00
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@ -12,6 +12,8 @@ title: Mathematical Proof Study Notes
## *Hammack*, 25 Jan 2014 ## *Hammack*, 25 Jan 2014
### Ch 1, Sets
All of Mathematics can be described with *sets*. All of Mathematics can be described with *sets*.
*set* *set*
@ -260,3 +262,137 @@ for at least one set $A_{\alpha}$ with } \alpha \in I \}\\
\bigcap_{\alpha \in I} A_{\alpha} &= \{ x : x \in A_{\alpha} \text { \bigcap_{\alpha \in I} A_{\alpha} &= \{ x : x \in A_{\alpha} \text {
for every set $A_{\alpha}$ with } \alpha \in I \}\\ for every set $A_{\alpha}$ with } \alpha \in I \}\\
\end{align}$$ \end{align}$$
* * * *
**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.