More notes

math/proof/notes.page

@@ -12,6 +12,8 @@ title: Mathematical Proof Study Notes
 
 ## *Hammack*, 25 Jan 2014
 
+### Ch 1, Sets
+
 All of Mathematics can be described with *sets*.
 
 *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 {
 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.