Induction and Recursion CMPS/MATH 2170: Discrete Mathematics - - PowerPoint PPT Presentation

Induction and Recursion

CMPS/MATH 2170: Discrete Mathematics

Outline

• Mathematical induction (5.1)
• Strong induction (5.2)
• Recursive definitions (5.3)
• Recurrence Relations (8.1)

Principle of Mathematical Induction

• Want to know if we can reach every step on a

infinite ladder

• Suppose we know two things
• We can reach the first rung of the ladder
• If we can reach a particular rung of the

ladder, then we can reach the next rung

• Can we conclude that we can reach every rung?

Mathematical Induction

• Want to show: ∀𝑜 ∈ ℤ%: 𝑄 𝑜

Proof by induction on 𝑜

• Base case: verify that 𝑄(1) is true
• Inductive step: show that 𝑄 𝑙 → 𝑄 𝑙 + 1 for any 𝑙 ∈ ℤ%

Inductive hypothesis: Assume 𝑄(𝑙) is true Want to prove 𝑄 𝑙 + 1 is true

Examples of Mathematical Induction

Ex.1: Sums of Arithmetic Progressions Ex.2: Sums of Geometric Progressions ∀𝑜 ∈ ℤ%: / 𝑗

1 234

= 𝑜(𝑜 + 1) 2 ∀𝑜 ∈ ℕ: / 𝑑2

1 239

= 𝑑1%4 − 1 𝑑 − 1 where 𝑑 ≠ 1

More about Mathematical Induction

• Why Mathematical Induction is valid?
• Implied by the Weak Ordering Property:

“Every nonempty subset of ℤ% has a least element”

• Pros
• Can be used to prove a wide variety of “forall” conjectures
• Easy to follow
• Cons
• Cannot be used to find new theorems
• Lack of insight

𝑄(1) ∀𝑙 ∈ ℤ%: 𝑄 𝑙 → 𝑄 𝑙 + 1 ∀𝑜 ∈ ℤ%: 𝑄 𝑜

∴

More Examples

• Ex. 3: Prove 𝑜= < 21 for all integers 𝑜 > 4
• Ex. 4: Prove that a finite set with 𝑜 elements has 21 subsets
• Ex. 5: Prove that every amount of postage of 12 cents or more can be formed

using just 4-cent and 5-cent stamps.

Mathematical Induction

• Want to prove 𝑄 𝑜 is true for 𝑜 = 𝑐, 𝑐 + 1, 𝑐 + 2, … , where 𝑐 ∈ ℤ

Proof by induction on 𝑜:

• Base case: verify that 𝑄(𝑐) is true
• Inductive step: show that 𝑄 𝑙 → 𝑄 𝑙 + 1 for any 𝑙 = 𝑐, 𝑐 + 1, 𝑐 + 2, …

Outline

• Mathematical induction (5.1)
• Strong induction (5.2)
• Recursive definitions (5.3)
• Recurrence Relations (8.1)

Strong Induction

• Want to prove: ∀𝑜 ∈ ℤ%: 𝑄 𝑜

Proof by (weak) induction on 𝑜:

• Base case: verify that 𝑄(1) is true
• Inductive step: show that 𝑄 𝑙 → 𝑄 𝑙 + 1 for any 𝑙 ∈ ℤ%

Proof by strong induction on 𝑜:

• Base case: verify that 𝑄(1) is true
• Inductive step: show that [𝑄 1 ∧ 𝑄 2 ∧ … ∧ 𝑄 𝑙 ] → 𝑄 𝑙 + 1 for any 𝑙 ∈ ℤ%

Strong Induction

• A more general form of strong induction
• Base step: verify that 𝑄 𝑐 , 𝑄 𝑐 + 1 , … 𝑄(𝑐 + 𝑘) are true
• Inductive step: Assume [𝑄 𝑐 ∧ 𝑄 𝑐 + 1 ∧ … ∧ 𝑄 𝑙 ] is true, prove 𝑄 𝑙 + 1 is

true for every integer 𝑙 ≥ 𝑐 + 𝑘

Examples of Strong Induction

Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, … 𝑔

9 = 0, 𝑔 4 = 1,

𝑔

1 = 𝑔 1K4 + 𝑔 1K=, 𝑜 ≥ 2

This is called a recursive definition Ex.1: 𝑔

1 ≤ 21 for all 𝑜 ≥ 0

• Ex. 2: 𝑔

1 > 𝛽1K= for any 𝑜 ≥ 3 where 𝛽 = (1 +

5

• )/2

wikipedia

Initial conditions Recurrence relation

Outline

• Mathematical induction (5.1)
• Strong induction (5.2)
• Recursive definitions (5.3)
• Recurrence Relations (8.1)

Recursively Defined Sequences

• Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, …

𝑔

9 = 0, 𝑔 4 = 1,

𝑔

1 = 𝑔 1K4 + 𝑔 1K=, 𝑜 ≥ 2

• A sequence of powers of 2: 1, 2, 4, 8, 16, 32 …

An explicit formula: 𝑏1 = 21, 𝑜 ≥ 0 A recursive definition: 𝑏9 = 29, 𝑏1 = 2𝑏1K4, 𝑜 ≥ 1

Recurrence relation Recurrence relation Initial conditions Initial condition

Recursively Defined Functions

• A recursive definition of 𝑔: ℕ → 𝐵, ℕ = {0,1,2, 3 … }
• Base step: specify 𝑔 0
• Recursive step: specify 𝑔 𝑜 in terms of 𝑔 0 , 𝑔 1 , … , 𝑔(𝑜 − 1), for any 𝑜 ≥ 1
• Ex.1: Give a recursive definition of 𝑔 𝑜 = 𝑜!
• Ex.2: Give a recursive definition of 𝑔 𝑜 = ∑

𝑏2,

1 239

where 𝑏2 2∈ℕ is a given sequence

Recursively Defined Sets

• Consider a set 𝑇 ⊆ ℤ% recursively defined by
• Base step: 3 ∈ 𝑇
• Recursive step: if 𝑦 ∈ 𝑇 and 𝑧 ∈ 𝑇, then 𝑦 + 𝑧 ∈ 𝑇
• Show that the set 𝑇 is the set of all positive integers divided by 3
• 𝐵 = 𝑜 ∈ ℤ%: 3 𝑜
• 𝐵 ⊆ 𝑇: prove by mathematical induction
• 𝑇 ⊆ 𝐵: prove by structural induction

Structural induction

• Want to prove a result for a recursively defined set
• Base step: show that the result holds for all base cases in the recursive

definition

• Inductive step: show that if the statement is true for each of the elements used

to construct new elements in the recursive step, the result holds for the new elements

Outline

• Mathematical induction (5.1)
• Strong induction (5.2)
• Recursive definitions (5.3)
• Recurrence Relations (8.1)

The Tower of Hanoi

Task: Move the stack of disks from peg 1 to peg 3 subject to the following rules:

• Move one disk at a time
• Only the uppermost disk on a stack can be moved
• No disk can be placed on top of a smaller disk

Question: how many steps are needed?

The Tower of Hanoi

By André Karwath aka Aka - Own work, CC BY-SA 2.5, https://commons.wikimedia.org/w/index.php?curid=85401

The Tower of Hanoi

• 𝐼1 - number of moves needed to solve the

Tower of Hanoi with 𝑜 disks

• Divide & Conquer

𝐼4 = 1 𝐼1 = 2𝐼1K4 + 1, 𝑜 ≥ 2

• How to find an explicit formula for 𝐼1?
• Expand the recurrence iteratively and then make

a conjecture: 𝐼1 = 21 − 1

• 264 − 1 seconds ≈ 585 billion years
• Proof by mathematical induction

𝑜 − 1 disks

𝐼1K4 𝐼1K4

1 move

Recurrence relation