Algorithms and Discrete Mathematics 20072008

1
Algorithms and Discrete Mathematics 2007/2008

• Lecture 8
• Revision

Ioannis Ivrissimtzis
30-Nov-2007
2
Revision

• Exponentials - Logarithms
• Properties of exponentials
• Properties of logarithms
• Big-O notation
• Definition
• Basic Big-O categories
• Big-O of polynomials
• Combinatorics
• Counting principles
• Permutations - combinations
• Binomial theorem
• Pascals triangle

3
Exponentials

• For a real number b?0 we define

Let b?0 be a real number and let n be a positive
integer. We define
Let b0 be a real number and let n be a positive
integer. We define as the n-th root of b. That
is, a real number x with the property
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Exponentials

• The exponential functions
• are everywhere positive
• For bgt1 they increase
• monotonically.

1
0
5
Exponentials

• Properties of the real exponents
• Let a,b,x,y be real numbers, with a,bgt0. We have

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Logarithms

• For real positive numbers x,b with b?1, the
  logarithm of x to the base
• b, written logbx is the unique real number y that
  satisfies byx.
• That is, if we raise b to the power of logbx we
  get x.

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Logarithms

• The logarithms are inverses
• of the exponentials.
• They are only defined on
• positive real numbers.

1
0
8
Logarithms

• Proposition 4.1 Let b,r,s be positive real
  numbers with b?1. We have
• These properties are related to properties of the
  exponents.

9
Revision

• Exponentials - Logarithms
• Properties of exponentials
• Properties of logarithms
• Big-O notation
• Definition
• Basic Big-O categories
• Big-O of polynomials
• Combinatorics
• Counting principles
• Permutations - combinations
• Binomial theorem
• Pascals triangle

10
The Big-O notation

• Let f and g be functions from the set of integers
  or the set of real
• numbers to the set of real numbers. We say that
  f(x) is O(g(x)) if there
• are constants C and k such that
• whenever x gt k. This is read as f(x) is
  big-oh of g(x).
• Rosen, p.180

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The Big-O notation

• Some orders that often occur in practice are

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Big-O of polynomials

• Proposition 5.1 Rosen, p.184 Any polynomial
  function is Big-O its
• leading degree without leading constant.

13
Revision

• Exponentials - Logarithms
• Properties of exponentials
• Properties of logarithms
• Big-O notation
• Definition
• Basic Big-O categories
• Big-O of polynomials
• Combinatorics
• Counting principles
• Permutations - combinations
• Binomial theorem
• Pascals triangle

14
Counting principles

• The product rule Rosen, p.336 Suppose that a
  procedure can be
• broken down into a sequence of two tasks. If
  there are n1 ways to do
• the first task and for each of these ways of
  doing the first task, there are
• n2 ways to do the second task, then there are
  n1n2 ways to do the
• procedure.

The sum rule Rosen, 338 If a task can be done
either in one of n1 ways or in one of n2 ways,
where none of the set of n1 ways is the same as
any of the set of n2 ways, then there are n1n2
ways to do the task.
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Counting principles

• Inclusion-exclusion principle Rosen, p.342 Let
  A1, A2 be finite sets,
• we have

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Permutations
Definition 6.2 Rosen, p.355 An ordered
arrangement of r elements of a set is called an
r-permutation.

• Theorem 6.1 Rosen, p.356 If n and r are
  integers with 1 r n, then
• there are
• r-permutations of a set with n distinct elements.

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Combinations

• Definition 6.4 Rosen, p.355 An r-combination
  of elements of a set is
• an unordered selection of r elements from the
  set.
• Thus, an r-combination is simply a subset with r
  elements.

Theorem 6.2 Rosen, p.358 The number of
r-combinations of a set with n elements, where n
and r are integers with 0 r n, equals
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Binomial theorem

• The Binomial Theorem Rosen, p.363 Let x and y
  be variables, and
• let n be a nonnegative integer. Then

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Pascals triangle
By Pascals identity
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Pascals triangle