Mathematical Induction and Other Proof Methods

(数学的帰納法などの証明方法)

Discrete Mathematics I

14th lecture, January 15, 2019

http://www.sw.it.aoyama.ac.jp/2018/Math1/lecture14.html

Martin J. Dürst

AGU

© 2006-19 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

Remaining Schedule

About makeup classes: The material in the makeup class is part of the final exam. If you have another makeup class at the same time, please inform the teacher as soon as possible.

補講について: 補講の内容は期末試験の対象。補講が別の補講とぶつかる場合には事前に申し出ること。

 

Questions about Final Exam

 

Summary of Last Lecture

 

Previous Homework

Draw the Hasse diagram of a Boolean algebra of order 4 (16 elements).

Sorry, it was removed! :)

 

Digit Sum and Digital Root

Digit sum: Sum of all of the digits of a number

Digital root: Single-digit result of repeatedly calculating the digit sum

Example in base 10:
The digit sum of 1839275 is 1+8+3+9+2+7+5 = 35
The digit sum of 35 is 3+5 = 8
The digital root of 1839275 is 8

Example in base 16:
The digit sum of A8FB is A+8+F+B (10+8+15+11) = (44) 2C
The digit sum of 2C is 2+C (2+12) = (14) E
The digital root of A8FB is E

 

Application of Congruence: Casting out Nines

 

About the Handout

 

Importance of Proofs

 

How Detailled Should a Proof Be?

 

Proof Methods

 

Proofs and Symbolic Logic

(S is the the theorem to be proven, expressed as a proposition or predicate)

 

Deduction and Induction

 

The Two steps of Mathematical Induction

S(0) ∧ (∀k∈ℕ: S(k) → S(k+1)) ⇒ (∀n∈ℕ: S(n))

  1. Base case (basis (step)): proof of S(0)
  2. Inductive step (induction, inductive case): proof of ∀k∈ℕ: S(k) → S(k+1)
    1. (inductive) Assumption: clearly state S(k)
    2. Actual proof of inductive step

      Method: Formula manipulation so that the assumption can be used

 

Simple Example of Mathematical Induction

Look at the following equations:

= 0 (adding up no numbers results in 0)

1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

Express the general rule contained in the above additions as a hypothesis.

Prove the hypothesis using Mathematical induction.

 

Hypothesis

Proof

1. Basis:

Prove the property for n = 0: ∑0i=1 2i-1 = 0 = 02

2. Induction:

a. Inductive assumption: Assume that the property is true for some k≥0: ∑ki=1 2i-1 = k2

b. Show that the property is true for k + 1:
(k+1)i=1 2i-1 = (k+1)2

[start with right side]
(k+1)2 [expansion]
= k2 + 2k + 1 [arithmetic]
= k2 + 2k + 2 - 1 [arithmetic]
= k2 + 2(k+1) - 1 [use assumption]
= ∑ki=12i-1 + 2(k+1) - 1 [property of ∑]
= ∑k+1i=1 2i-1 Q.E.D.

 

Variations of Mathematical Induction

 

Example of Structural Induction

 

Proof of the Relation between the Number of Nodes and Leaves in a Binary Tree

We start with a very small tree consisting only of the root, and grow it step by step. We can create any shape of binary tree this way.

  1. Base case: In a tree with only the root node, n=1 and l=1, therefore n = 2l-1 is correct.
  2. Inductive step: Grow the tree one step by replacing a leaf with an internal node with two leaves.
    Denote the number of nodes before growing by n, the number of leaves before growing by l, the number of nodes after growth by n', and the number of leaves after growth by l'. We need to prove n' = 2l'-1.
    1. Inductive assumption: n = 2l-1
    2. In one growth step, the number of nodes increases by two: n'=n+2 (1)
      In one growth step, the number of leaves increases by two but is reduced by one: l'=l+2-1=l+1; l = l'-1 (2)
      n' [(1)]
      = n+2 [assumption]
      = 2l-1+2 [(2)]
      = 2(l'-1)-1+2 [arithmetic]
      = 2l'-1 Q.E.D.

 

Homework

(no need to submit)

  1. Answer the question on the slide "Application of Congruence: Casting out Nines"
  2. Find out the problem in the following proof:

    Theorem: All n lines on a plane that are not parallel to each other will cross in a single point.

    Proof:

    1. Base case: Obviously true for n=2
    2. Induction:
      1. Assumption: k lines cross in a single point.
      2. For k+1 lines, both the first k lines and the last k lines will cross in a single point, and this point will have to be the same because it is shared by k-1 lines.
  3. Read the handout
  4. Find a question regarding past examinations that you can ask in the next lecture.

 

Glossary

digit
数字、桁
digit sum
数字和
digital root
数字根
casting out nines
九去法
(formal) language theory
言語理論
automata theory
オートマトン理論
proof
証明
to prove
証明する
data structure
データ構造
intuition
直感
test case
テスト・ケース
deductive proof
演繹的証明
inductive proof
帰納的証明
proof by contradiction
背理法
proof by counterexample
反例による証明
proof about sets
集合についての証明
proof by enumeration
列挙による「証明」
mathematical induction
数学的帰納法
structure
構造
loop
繰返し
base case
基底
inductive step
帰納
inductive assumption
(帰納の) 仮定
recursion
再帰
hypothesis
仮説
equation
方程式
consecutive
連続的な
odd (number)
奇数
inductive assumption
(帰納の) 仮定
strong induction
完全帰納法 (または累積帰納法)
structural induction
構造的帰納法
binary tree
二分木
node (of a tree/graph)
leaf (of a tree)
directed graph
有効グラフ
cycle (of a graph)
閉路
parent (in a tree)
child (in a tree)
root (of a tree)
internal node
内部節
parallel
平行