Algorithms and Discrete Mathematics 20062007

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Algorithms and Discrete Mathematics 2006/2007

• Lecture 10
• Fundamentals of Discrete Mathematics
• Proof Methods II

Ioannis Ivrissimtzis
08-Dec-2006
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Overview of the lecture

• What is a proof ?
• Methods and principles for mathematical proofs
• Direct proofs
• Reductio ad absurdum (proof by contradiction)
• Mathematical induction
• Pigeonhole principle
• What is not a proof

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Mathematical induction

• Principle of mathematical induction
• If a set of positive integer numbers contains the
  integer 1, and also
• contains the integer n1 whenever it contains the
  integer n, then it
• contains all the positive integer numbers.

The principle of mathematical induction says that
if (i) The set contains 1 (induction
base). (ii) If the set contains n (induction
hypothesis), then it also contains n1. then
the set contains all the positive integers.
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Mathematical induction

• The logic behind the principle of mathematical
  induction is that
• The set contains 1. (induction base)
• It contains 2 because it contains 1.
• It contains 3 because it contains 2.
• It contains 4 because it contains 3.
• It contains n1 because it contains n.
• It contains n2 because it contains n1.

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Mathematical induction

• The mathematical induction can be used to prove
  statements of the
• form
• The proposition P(n) is true for every positive
  integer n.
• Indeed, it suffices to prove that
• (i) P(1) is true (induction base)
• (ii) If P(n) is true (induction hypothesis),
  then P(n1) is also true.
• Or, in other words, that P(n)
  implies P(n1).

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Mathematical induction

• In a variant of the principle of mathematical
  induction
• If
• (i) P(n0) is true, for some positive integer n0
  (induction base)
• (ii) If P(n) is true (induction hypothesis),
  then P(n1) is also true.
• then,
• P(n) is true for every positive integer n n0.

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Mathematical induction

• Example 10.1 For any integer n 1 we have
• Proof For n 1 the equation holds as
• We assume that the equation holds for k, that is,
  the induction
• hypothesis is

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Mathematical induction

• Then we have,
• That is, the equation also holds for k1, and the
  inductive proof is
• complete.
• ?

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Mathematical induction

• Example 10.2 The Fibonacci numbers are defined
  as
• The first few Fibonacci numbers are 0, 1, 1,
  2, 3, 5, 8, 13, 21,
• Show that
• for all n 1.

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Mathematical induction

• Proof
• Induction base
• For n 1,
• and
• Therefore, the proposition is true for n 1.

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Mathematical induction

• Induction step
• Let the proposition be true for an arbitrary but
  fixed positive integer k.
• That is, the induction hypothesis is
• We have to show that the proposition is also true
  for k1, that is, we
• have to show that

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Mathematical induction

• Indeed,
• and the inductive proof is complete.

Fibonacci definition Induction
hypothesis Fibonacci definition
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Overview of the lecture

• What is a proof ?
• Methods and principles for mathematical proofs
• Direct proofs
• Reductio ad absurdum (proof by contradiction)
• Mathematical induction
• Pigeonhole principle
• What is not a proof

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Pigeonhole principle

• Pigeonhole principle
• If n1 pigeons fly to n holes, there must be at
  least one hole containing
• at least two pigeons.
• In a more mathematical language, if we map a set
  A with n1
• elements on a set B with n elements, then there
  is at least one element
• of B which is the image of at least two elements
  of A.

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Pigeonhole principle

• Example 10.3 Let a, b, c be integers. We can
  choose two of them
• whose sum is even.
• Proof
• Apply the pigeonhole principle with the sets of
  odd and even integers
• as the two holes, and the a, b, c as the three
  pigeons
• Two of the a, b, c have the same parity, that is,
  they are both even, or
• both odd.
• The sum of two even numbers, or the sum of two
  odd numbers is even.
• ?

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Pigeonhole principle

• Example 10.4 Five points are given on, or inside
  a unit square. Show
• that there is a pair of points whose distance is
  at most .
• Proof Split the unit square into 4 squares with
  edge 1/2 as in the
• figure. By the pigeonhole principle, two of the
  points are on, or inside
• one of the four smaller squares.

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Pigeonhole principle
Their distance is at most the maximum distance
between two points of that square, that is, at
most the diameter of the square. By the
Pythagorean theorem, the diameter of the small
square is ?
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Overview of the lecture

• What is a proof ?
• Methods and principles for mathematical proofs
• Direct proofs
• Reductio ad absurdum (proof by contradiction)
• Mathematical induction
• Pigeonhole principle
• What is not a proof

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What is not a proof

• You can not prove a statement of the form
• The proposition P(n) is true for every
  positive integer n
• by just doing the computations for several values
  of n.
• Even if we use a computer and verify the P(n) is
  true for millions or
• billions of different values of n, still this is
  not a proof.

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What is not a proof

• Even though they do not constitute a proof, the
  computation of special
• cases (usually corresponding to small values of
  n) can be useful.
• They can give an insight into the problem,
  sometimes even the idea for
• a proof.
• They might give a counterexample. That is,
  finding just one positive
• integer n0 such that P(n0) is false, it suffices
  to show that the claim
• P(n) is true for every positive integer n
• is false.

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What is not a proof

• Example 10.5 The numbers
• are all prime numbers. Fermat (a 17th century
  mathematician)
• conjectured that
• is prime for every positive integer n.

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What is not a proof

• About 100 years later, Euler computed that
• is not a prime.
• This counterexample disproved Fermats
  conjecture.