Mathematical Induction

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Mathematical Induction

• An introduction to proofs

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NC Standard Course of Study

• Competency Goal 3 The learner will describe and
  use recursively-defined relationships to solve
  problems.
• Objective 3.01 Use recursion to model and solve
  problems.
• Find the sum of a finite sequence.
• Find the sum of an infinite sequence.
• Determine whether a given series converges or
  diverges.
• Write explicit definitions using iterative
  processes, including finite differences and
  arithmetic and geometric formulas.
• Verify an explicit definition with inductive
  proof.

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How To Verify Patterns

• This lesson is concerned with the way that
  certain kinds of patterns are verified.
• Because the prediction made by patterns can be
  erroneous and can result in the expenditure of
  unnecessary effort and money, it is necessary
  that they be as accurate as possible.
• The reasoning method used to verify some patterns
  is mathematical induction.

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Mathematical Induction

• This method is used to prove that certain types
  of discrete patterns continue.
• For example, with the cake division by the
  cut-and-choose method, the method can continue
  indefinitely.
• Initially, it began by looking at a situation
  with two people and then the method was extended
  to 3, 4 and more people.
• In each example, the method requires that all but
  one person cut and the last person choose.

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Extending the Method

• When considering four people, the method requires
  that three of the people divide their piece into
  four pieces, and that the fourth person choose
  the three pieces they want.
• The cutters must feel that they are left with
  three portions that are each one-fourth of their
  original share, which is at least one-third of
  the cake.

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The Fourth Person

• Although the fourth person may not feel that each
  of the three portions is at least one-third of
  the cake, s/he must feel that the total value of
  the three portions is 1.
• Suppose the fourth person assigns values p1, p2,
  and p3 to the three portions.
• Then p1 p2 p3 1.

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The Fourth Person (contd)

• Because the fourth person is given the first
  choice of a portion from each of the original
  three people, s/he will place a value of at least
  
• on the resulting portion.
• Accordingly, or one-fourth of the
  entire cake.

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Extending the Method

• Does this method work with 5 people, 6 people, 7
  people, 8 people, ?
• Yes, it does!
• The fact that it works is based on the
  mathematical principle of induction
• Mathematical induction generalizes this pattern
  of solutions by proving that it is always
  possible to extend the solution to a group that
  is one larger than the previous. The
  generalization is achieved by using a variable
  rather than a specific number.

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Dividing the Cake

• Suppose you know how to divide a cake fairly
  among k people. You need to show that it is also
  possible to divide a cake fairly among k 1
  people.
• This shows that the two-person solution can be
  extended to more than two people.

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The proof

• By applying the assumption that k people can
  fairly divide the cake, then each person must
  divide their cake into k 1 portions, that each
  feels are equal.
• The k 1st person then selects one portion from
  each.
• Then it must be proved that this results in a
  share of at least 1/ (k 1) for each of the k
  1 people.

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The Proof (contd)

• Of the k k people who cut the cake, each should
  feel that each portion is 1/(k1) of at least 1/k
  of the cake.
• Multiplying those gives 1/ (( k 1 )k).
• Each person gets to keep k of the k 1 portions,
  which gives a total value of at least k
  (1/((k1)k) 1/ (k1).

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The Proof (contd)

• Although the chooser may not feel that all of the
  original k portions are at least 1/k of the cake,
  s/he must feel that the total value is 1.
• If the person assigns values of p1, p2, ..pk to
  the k pieces, then p1 p2 pk 1.

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The Proof (contd)

• Because the chooser chooses first s/he is willing
  to place a value of at least
• on the resulting portion.
• By factoring out the 1/(k1) and since p1 p2
  pk 1 then each person gets 1/(k1) of the
  cake.

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Using Mathematical Induction

• The proof is complete since it shows that
  whenever a cake is divided fairly among k people,
  it can also be divided fairly among k 1 people.
• Mathematical induction is frequently used to
  verify that an observed formula always works.

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An Example of Induction

• Luis and Britt are investigating the number of
  handshakes that will be made by a group of people
  if each person shakes hands with every other
  person.
• Luis notes that if there is only one person, no
  handshakes are possible and that if there are two
  people, only one handshake is possible.

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Example (contd)

• This information can be represented either by a
  graph or a table as shown below

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Number of People in Group Number of Handshakes
1 0
2 1
3 3
1
2
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Practice Problems

• To use mathematical induction, you must be able
  to use symbols to express numeric patterns. Some
  of the expressions you write in this exercise
  will be used in the mathematical induction proof.
• a. If there are three people in a group and
  another person joins the group, there will be
  four people in the group. If a person leaves the
  original group of three, there will be two.
  Write expressions for the number of people if
  there are k people in a group and another person
  joins. Do the same if a person leaves the group
  of k people.

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Practice Problems (contd)

• Repeat this exercise for a group of k 1 people,
  and then for a group of 2k people.
• Draw a graph like Britts and a table like
  Luiss.
• a. Add another vertex to the graph to represent
  a fourth person, and draw segments to represent
  the additional handshakes that will result if the
  group grows to four people. Determine the number
  of handshakes in a group of four by adding the
  number of new handshakes to the number for a
  group of three given in the table. Write in your
  table the total number of handshakes for a group
  of four people.

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Practice Problems (contd)

• b. Add a fifth vertex to represent a fifth
  person, and draw segments to represent the
  additional handshakes. Add the number of new
  handshakes to the number for a group of 4 given
  in the table. Write in your table the total
  number of handshakes for a group of 5 people.

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Practice Problems (contd)

• a. Suppose that there are seven people in a group
  and each of them has shaken hands with every
  other person. If an eighth person enters the
  group, how many additional handshakes must be
  made?
• b. Suppose that there are k people in a group
  and each of them has shaken hands with every
  other person. If a new person enters the group,
  how many additional handshakes must be made?

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Practice Problems (contd)

• After studying the data for a while, Britt
  wonders whether the number of handshakes in a
  group can be found by multiplying the number of
  people in the group by the number that is 1 less
  than that and dividing this product by 2.
• a. If her guess is correct, how many handshakes
  would there be in a group of 10 people?

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Practice Problems (contd)

• Write an expression for the number of handshakes
  based on Britts guess if there are k people in a
  group. Do the same for a group of k 1 people.

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Recurrence Relations

• Britts formula, if correct, is sometimes known
  as a solution of the recurrence relation.
• A recurrence relation is a verbal or symbolic
  statement that describes how one number in a list
  can be derived from the previous number.

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Recurrence Relations (contd)

• One of the advantages of a recurrence relation is
  that it allows you to determine the number of
  handshakes in a group without using the number of
  handshakes in a smaller group.
• Let Hn represent the number of handshakes in a
  group of n people, what is the recurrence
  relation that expresses the relationship between
  Hn and Hn-1? Write the recurrence relation that
  expresses the relationship between Hn1 and Hn.

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Checking Britts Guess

• To prove that Britts guess is correct, show that
  whenever the solution is known to work, it is
  possible to extend it to a group that is 1
  larger.
• In other word, whenever the conjecture works for
  a group of k people, it will also work for a
  group of k 1 people.

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Practice Problems (contd)

1. Assume that Britts formula works for a group of
   k people, and write the formula for such a group.
2. You need to show that Britts formula works for a
   group of k 1 people. Write the formula for k
   1 people.
3. If an additional person enters a group of k
   people, how many new handshakes are necessary?

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Total Number of Handshakes

• An expression for the total number of handshakes
  in a group of k 1 people can be found by adding
  the expression for the number of handshakes in a
  group of k people (part c) to the number of new
  handshakes (part e)

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Proof

• You can conclude that Britts formula will always
  work if this expression matches the one in part
  d. Use algebra to transform the expression until
  it matches the one you wrote in part d.

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Practice Problems (contd)

• Although Britts formula is for the number of
  handshakes in a group of people, it could also
  represent the number of potential two-party
  conflicts in a group.
• a. Use the formula to compare the number of
  potential conflicts when the size of a group
  doubles. Does the number of potential disputes
  also double?

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Practice Problems (contd)

• b. Why do the results of Exercise 4 suggest that
  some of the costs associated with government,
  such as that of maintaining a police force, may
  outpace the growth of a population?

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Beginning a Proof

• In Exercises 1-4 you supplied several of the
  steps of the mathematical induction proof that
  began in the lesson. In Exercise 6, you will
  again supply many of the steps of the induction
  process, which requires a number of preliminary
  steps leading to the guessing of a formula, which
  must be proved.

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Preliminary Steps

• The preliminary steps are summarized here
• Organize a table of data for several small
  values. For example, how many ways of voting are
  there with 1, 2, 3, or 4 choices on the ballot?
• Investigate the problem and the data to describe
  the pattern of the data with a recurrence
  relation. For example, how many ways of voting
  are added when another choice is placed on the
  ballot?

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Prelim. Steps (contd)

1. Make up a formula that predicts the outcome for a
   collection of k items. For example, what is a
   formula that predicts the number of ways of
   voting when there are k choices on the ballot?
2. Verify that your formula works for the small
   values you have tabulated.

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Practice Problems (contd)

• 6. Lets look at an approval voting situation.
  Lets use mathematical induction to verify that a
  suspected formula for the number of ways of
  voting under the approval system when there are n
  choices on the ballot is indeed correct.

Number of Choices on the Ballot 1 2 3 4
Number of Possible Ways of Voting 2 4 8 16