Mathematical Preliminaries

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Mathematical Preliminaries
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Sets

• A set is a collection of distinguishable
  elements.
• elements are either a primitive or another set.
• no duplicates
• Terminology
• base type
• primitive element

1, 4 5, 2, 3 , 22 P 5, 16, 20
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Set Terminology

• Assume the following definitions

A 2, 3, 5 B 5, 10
x x is a positive integer x ? B P( x )
x ? A x ? A ? ??A??
A ? B A ? B A ? B A ? B A B A ? B A ? B A - B
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Powerset

• The powerset of a set A is the set of all
  possible subsets for A.

A 2, 3, 5
A 2, 3, 5 The powerset of A is ?, 2 ,
3 , 5 , 2, 3, 2, 5, 3, 5, 2, 3, 5
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Linear Order

• A linear order has the following properties

• For every a, b, and c in a set
• a lt b, a b, or a gt b
• if a lt b and b lt c, then a lt c.

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Factorial Function

• Grows quickly as n becomes larger.
• Time consuming.
• Stirling's approximation

n! n ? ( n 1 ) ? ? ? 2 ? 1
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Permutations

• The arrangement of the elements of a sequence.

????????? ???

• How many permutations are there for a sequence of
  length n?
• What are all the permutations for the above
  sequence?

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Floor and Ceiling

• The floor of a real value x is the greatest
  integer less than or equal to x.

y ? x ?

• The ceiling of a real value x is the smallest
  integer greater than or equal to x.

y ? x ?
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Modulus Operator

• Returns the remainder of integer division.
• Alternate

n mod n is the integer r s.t. n qm
r for q an integer and 0 lt r lt m.
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Logarithms

• If
• then
• The square of a logarithm log2n
• The logarithm of a logarithm log log n

x logb y
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Logarithms (cont)

• Logarithms have the following properties

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Summations

• The sum for some function over a range of
  parameter values.
• Which is the same as

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Summations (cont)
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Summations (cont)
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Summations (cont)

• Special cases of the last equation are

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Summations (cont)

• A corollary of the equation
• is

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Recurrences

• recurrence relation a function which is defined
  in terms of a smaller instance of itself.

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Recurrences (cont)

• alternate form
• base case stops the recursion
• expanding

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Proof by Contradiction

• We can disprove a theorem or statement by
  counterexample.
• But, we can not prove by example.
• We can prove by contradiction.

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Proof by Contradiction (cont)

• Technique
• given a theorem, assume it is false
• find a logical contradiction
• If logic is correct, the assumption must be wrong
  and the theorem correct.

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Prove by Contradiction Examples

• There is no largest integer.
• Given any integer n, if n2 is even, then n is
  even.

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Proof by Mathematical Induction

• Suppose T is a theorem to be proved which can be
  expressed in terms of a positive integer n.
• Then T is true for any value of n gt c, if the
  following conditions are true
• base case if T holds for n c, and
• induction step if T holds for n - 1, then T
  holds for n.

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Proof by Induction (cont)

• Three steps to an induction proof
• prove the base case
• state the induction hypothesis
• use the induction hypothesis
• If we can show that T is true for n 1, then T
  is true for values gt c.