Math 365

  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
  33. 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
  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
    • 19 add ones
    • 130 add tens
    • +400 add hundreds
    • 549
  37. Left to Right Algorithm
    • 458
    • +832
    • 1200 (400+800)
    • 80 (50+30)
    • + 10 (8+2)
    • 1200
    • + 90 (80+10)
    • 1290
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Math 365
Math 365 Exam Cards