# Math 365

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1. Polya's Four-Step Problem-Solving Process
• 1. Understand the problem
• 2. Devise a plan
• 3. Carry out the plan
• 4. Looking back
2. 1. Understanding the problem
Can you state the problem in your own words
3. 2. Devise a plan
Look for a pattern
4. 3. Carrying out the plan
Check each step of the plan as you proceed
5. 4. Looking back
check the results in the original problem
6. Conjecture
a statement throught to be true, but not proven
7. Counterexample
example that contradicts the conjecture, shows the conjecture false
8. Arithmetic Sequence
an= a1+ d(n-1)
9. Geometric Sequence
an = a1* r(n-1)
10. Recursive Sequence
Ex: a1=2, a2=3, an=3an-2-an-1, for natural #n>2

must have all 3 parts or will be wrong
11. In logic, a statement is a sentence that is
either T or F
12. 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
13. 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
14. 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*
15. Place Value
assigns a value of a digit depending upon its placement in a numeral
16. Definition of an
if a is any # and n e N, then an= a*a*...*a

Ex: 23= 2*2*2=8
17. Mayan Numeration System
• a0=1
• a1=20
• a2=20*18=360
• a3=202*18=7200...etc
18. Dozen: Base 12
• gross = dozen dozen
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E
19. Sets P & Q are in one-to-one 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
20. 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
21. Two sets A & B are equivalent A~B
iff there exists a 1-1 correspondence btwn the two sets.
22. The cardinal # of a set A, n(A):
indicates the # of elelments in set A
23. A set is finite
if its cardinal number is a whole #
24. The complement of a set A, written Ac:
is the set of all elements in the universal set U that are not in A
25. 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
26. Inequalities
are an application of set concepts
27. "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.
28. How many subsets does a finite set have?
it has 2n(A)subsets
29. How many proper subsets does a finite set have?
it has 2n(A)-1
30. Set complement of A relative to B: B-A = {x|x e B and x e A}
meaning in B but not in A
31. 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)
32. 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
• 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
34. Def of Subtraction of W
for any a, b e W, such that a > b, a-b is a unique c eW such that a=b+c
35. 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
36. Expanded Algorrithm:
• 125
• 345
• + 79