
Polya's FourStep ProblemSolving Process
 1. Understand the problem
 2. Devise a plan
 3. Carry out the plan
 4. Looking back

1. Understanding the problem
Can you state the problem in your own words

2. Devise a plan
Look for a pattern

3. Carrying out the plan
Check each step of the plan as you proceed

4. Looking back
check the results in the original problem

Conjecture
a statement throught to be true, but not proven

Counterexample
example that contradicts the conjecture, shows the conjecture false

Arithmetic Sequence
a_{n}= a_{1}+ d(n1)

Geometric Sequence
a_{n} = a_{1}* r^{(n1)}

Recursive Sequence
Ex: a_{1}=2, a_{2}=3, a_{n}=3a_{n2}a_{n1}, for natural #n>2
must have all 3 parts or will be wrong

In logic, a statement is a sentence that is
either T or F

The negation of a statement is a statement w the opposite true value of the given statement
 Be careful w quantifiers:
 Universal: all, every, & no refers to each & every element in a set
Existential: some, there exists at least one refers to one or more or passible all elements in a set

Truth tables
 p^q (p and q)  if both are T then its T
 pVq (p or q)  if both are F then its F

Truth Tables
 Conditional Statements:
 p > q (if p then q)
 Converse:
 q > p (if q then p)
 Inverse:
 ~p > ~q (if not p then not q)
 Contrapositive:
 ~q > ~p (if not q them not p)
 Biconditional:
 p <> q (p iff q)
* If 1st is T & 2nd is F then its F*

Place Value
assigns a value of a digit depending upon its placement in a numeral

Definition of a^{n}
if a is any # and n e N, then a^{n}= a*a*...*a
Ex: 2^{3}= 2*2*2=8

Mayan Numeration System
 a^{0}=1
 a^{1}=20
 a^{2}=20*18=360
 a^{3}=20^{2}*18=7200...etc

Dozen: Base 12
 gross = dozen dozen
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E

Sets P & Q are in onetoone correspondence
if elements of P and Q can be paired so that for each element of P there is exactly one element of Q, & for each element of Q there is exactly one element of P

Fundamental Counting Principle
If event M can occur in m ways, and after it has occurred, event N can occur in n ways, then event M followed by event N can occure in mn ways

Two sets A & B are equivalent A~B
iff there exists a 11 correspondence btwn the two sets.

The cardinal # of a set A, n(A):
indicates the # of elelments in set A

A set is finite
if its cardinal number is a whole #

The complement of a set A, written A^{c:}
is the set of all elements in the universal set U that are not in A

The empty set is a subset of everyset. Why?
 for any set A, either {}c A, or {} c A. Suppose{}c A, then there is some element in the empty set that is not in A, but because {} has no elements, it cannot have an element that is not in A.
 therefore {}c A

Inequalities
are an application of set concepts

"Less Than" using sets:
If A and B are finite sets then n(A) is less than n(B), written n(A)<n(B), if A is equicalent to a proper subset of B. So if n(A)=a & n(B)=b, then a<b. Similarly we define greater than: n(A)>n(B) or a>b, which is n(B)<n(A) or b<a, respectively.

How many subsets does a finite set have?
it has 2^{n(A)}subsets

How many proper subsets does a finite set have?
it has 2^{n(A)}1

Set complement of A relative to B: BA = {xx e B and x e A}
meaning in B but not in A

Def of addition of Whole #'s
Let A and B be disjoint ^{(A intercect B=0) }finite sets: If n(A)=a and n(B)=b, then a+b=n(A u B)

Def of Less Than:
for any a,b e W, a is less than b, written a<b, iff there exists a k e N such that a+k=b

Whole # Addition Properties
 Closure: if m,n e W, then m+n e W;
 Commutative: a+b = b+a
 Associative: (a+b)+c = a+(b+c)
 Unique Identity 0: a+0=0+a=a

Def of Subtraction of W
for any a, b e W, such that a > b, ab is a unique c eW such that a=b+c

The Number Line Model  adding & subtracting
 Start at zero facing the (+) direction
 Add means stay facing same direction
 Subtact means turn around
 (+) # means go forward
 () # means go backwards

Expanded Algorrithm:
 125
 345
 + 79
 19 add ones
 130 add tens
 +400 add hundreds
 549

Left to Right Algorithm
 458
 +832
 1200 (400+800)
 80 (50+30)
 + 10 (8+2)
 1200
 + 90 (80+10)
 1290

