snick  snack CPSC 121: Models of Computation 2011 Winter Term 1 Introduction to Induction Steve Wolfman 1

Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Single-Elimination Tournaments –Binary Trees A Pattern for Induction Induction on Numbers Next Lecture Notes 2

Quiz Notes You’re faced with a problem like “How many teams can play in a 100 round tournament?” or “How many teams can play in an n-round tournament?” You don’t know the answer. What can you do? Strategies: Try to clearly define terms (What’s a tournament? What’s a round?) Try a smaller number. 100 is hard, what about 1 or 2? Write the problem (or a small version) out as a story and figure out which parts of the story matter. 3

Learning Goals: Pre-Class By the start of class, you should be able to: –Convert sequences to and from explicit formulas that describe the sequence. –Convert sums to and from summation/“sigma” notation. –Convert products to and from product/“pi” notation. –Manipulate formulas in summation/product notation by adjusting their bounds, merging or splitting summations/products, and factoring out values. 4

Learning Goals: In-Class By the end of this unit, you should be able to: –Given a theorem to prove via induction and the insight into how the problem breaks down into one or more smaller problem(s), write out a complete proof strategy. –Prove naturally self-referential properties by induction. (That is, identify the “insight” and flesh out the strategy into a complete proof.) 5

Where We Are in The Big Stories Theory How do we model computational systems? Now: Developing a new proof technique that’s perfect for algorithms involving recursion (or iteration)... which is almost all interesting algorithms! Hardware How do we build devices to compute? Now: Taking a break in lecture. In lab, continuing to build toward the complete, working computer! (Although lab’s taking a break soon, too, for regular expressions.) 6

Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Single-Elimination Tournaments –Binary Trees A Pattern for Induction Induction on Numbers Next Lecture Notes 7

Single-Elimination Tournaments In each round teams play in pairs. Losing teams are eliminated. The tournament ends when only one team remains. 8

What is a Tournament? Let’s think like CPSC 110ers… A tournament is one of: 9

Single-Elimination Tournament Problem What’s the maximum number of teams in a 0-round single-elimination tournament? a.0 teams b.1 team c.2 teams d.3 teams e.None of these 10 Think: what does this correspond to in the data definition?

Single-Elimination Tournament Problem What’s the maximum number of teams in a 1-round single-elimination tournament? a.0 teams b.1 team c.2 teams d.3 teams e.None of these 11 A “one round tournament” has only one set of games. A “two round tournament” has two sets of games (finals and semi-finals).

Single-Elimination Tournament Problem What’s the maximum number of teams in a 2-round single-elimination tournament? a.0 teams b.1 team c.2 teams d.3 teams e.None of these 12 Would there be any “byes” in the tournament? A team with a “bye” gets to advance to the next round “for free”.

Single-Elimination Tournament Problem What’s the maximum number of teams in an n-round single-elimination tournament? a.n teams b.2n teams c.n 2 teams d.2 n teams e.None of these 13

Tournament Step-by-Step Problem 2: Prove your result for... n = 0 n = 1 n = 2 n =

Tournament Logic If at most 2 n-1 teams can play in an (n-1)-round tournament, then at most 2 n teams can play in an n-round tournament. If we want to prove this statement, which of the following techniques might we use? a.Antecedent assumption b.Witness proof c.WLOG d.Proof by cases e.None of these 15

The “insight into how the problem breaks down” A tournament of n rounds is made up of two tournaments of (n-1) rounds, like how the Stanley Cup Finals is set “on top of” the “east half” and “west half” tournaments. 16 One extra round: east winner vs. west winner

The “insight into how the problem breaks down” The n-round tournament with the maximum number of teams is made from two (n-1)-round tournaments with the maximum number of teams. 17

Completing (?) the Proof Theorem: if at most 2 n-1 teams can play in an (n-1)- round tournament, then at most 2 n teams can play in an n-round tournament. Proof: Assume at most 2 n-1 teams can play in an (n-1)- round tournament. An n-round tournament is two (n-1)-round tournaments where the winners play each other (since there must be a single champion). By assumption, each of these may have at most 2 n-1 teams. So, the overall tournament has at most 2*2 n-1 = 2 n teams. QED! 18

Are We Done? Here’s the logical structure of our theorem:  n  Z 0, Max(n-1,2 n-1 )  Max(n,2 n ). Does that prove  n  Z 0, Max(n,2 n ) ? a.Yes. b.No. c.I don’t know. 19 P.S. Is this really what we proved? Is our theorem true for all non-negative integers?

What More Do We Need? This parallels the self-referential variant of our tournament “data definition”:  n  Z 0, (n > 0)  Max(n-1,2 n-1 )  Max(n,2 n ). Why doesn’t this work for 0? What do we do about our base case of our data definition? 20

Completing (?) the Proof (again) Base Case Theorem: At most one team can play in a 0-round tournament. Proof: Every tournament must have one unique winner. A zero-round tournament has no games; so, it can only include one team: the winner. QED! 21 Everybody’s a winner. That’s so sweet

Now Are We Done? Here’s the logical structure of our theorems: Max(0,1).  n  Z 0, (n > 0)  Max(n-1,2 n-1 )  Max(n,2 n ). Do these prove  n  Z 0, Max(n,2 n ) ? a.Yes. b.No. c.I don’t know. 22

Step-by-Step? Here’s the logical structure of our theorems: Max(0,2 0 ).  n  Z 0, (n > 0)  Max(n-1,2 n-1 )  Max(n,2 n ). Do these prove Max(1,2 1 ) ? a.Yes. b.No. c.I don’t know. 23

Step-by-Step? Here’s the logical structure of our theorems: Max(0,2 0 ).  n  Z 0, (n > 0)  Max(n-1,2 n-1 )  Max(n,2 n ). Plus, we know Max(1,2 1 ). Do all of these prove Max(2,2 2 ) ? a.Yes. b.No. c.I don’t know. 24

Step-by-Step? Here’s the logical structure of our theorems: Max(0,2 0 ).  n  Z 0, (n > 0)  Max(n-1,2 n-1 )  Max(n,2 n ). Plus, we know Max(1,2 1 ) and Max(2,2 2 ). Do all of these prove Max(3,2 3 ) ? a.Yes. b.No. c.I don’t know. 25

Step-by-Step? Here’s the logical structure of our theorems: Max(0,2 0 ).  n  Z 0, (n > 0)  Max(n-1,2 n-1 )  Max(n,2 n ). From this, can we prove Max(n,2 n ) for any particular integer n ? a.Yes. b.No. c.I don’t know. 26

Tournament Proof Summary Theorem: 2 n teams can play in an n-round tournament. Proof: We proceed by induction on n. Base Case: Every tournament must have a single winner. A zero-round tournament has no games and so can only include one (that is, 2 0 ) team: the winner. So, 2 0 teams can play in a 0-round tournament.  Inductive Step (for n greater than 0: if at most 2 n-1 teams can play in an (n-1)-round tournament, then at most 2 n teams can play in an n-round tournament): WLOG, let n be an integer greater than 0. Assume at most 2 n-1 teams can play in an (n-1)-round tournament. An n-round tournament is two (n-1)-round tournaments where the winners play each other (since there must be a single champion). By assumption, each of these may have at most 2 n-1 teams. So, the overall tournament has at most 2*2 n-1 = 2 n teams.  This completes our induction proof. QED 27

Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Single-Elimination Tournaments –Binary Trees A Pattern for Induction Induction on Numbers Next Lecture Notes 28

“Binary Trees” A “binary tree” is either a leaf or a root “node” and two children, each of which is a binary tree. (Note: a slightly different definition is common, but less convenient.) 29 (They let us do some cool CS things... see CS110, 210, 221.)

Binary Tree Data Definition A binary-tree is one of: -A leaf -(make-node binary-tree binary-tree number)

Binary Trees’ Height A binary tree’s “height” is the maximum number of steps it takes to get from the root down to a node with only leaves as children. This one’s height is: a.0 b.1 c.2 d.3 e

Binary Trees’ Height Problem 1: What’s the maximum number of nodes in a binary tree of height n?

Binary Trees’ Height Problem 2: Prove your result

Binary Trees, Take Two Problem 1: Give an algorithm to determine the height of a binary tree. Problem 2: Prove that your algorithm is correct

Worked: Algorithm for Height Height(t): Is t a leaf? Yes: evaluate to -1 No: recursively find the height of each subtree h l and h r and evaluate to max(h l, h r )

Worked: Proof of Algorithm’s Correctness We proceed by induction on the size of the tree. Base case: On a leaf (the smallest tree), our algorithm gives a height of -1. We must go up one level to reach a node with leaves as children; so, this is correct. (Really, it’s just correct by definition.) Inductive case: Assume our algorithm works correctly on all trees smaller than a particular non-leaf tree t. t is not a leaf; so, our algorithm calculates the height of each subtree h l and h r. These heights are correct by assumption. To reach a node with only leaves as children, we go into either the left or right subtree. The length of the path through the left subtree is therefore h l +1, with the +1 for the step down into that subtree. Similarly, the right path is h r +1. The height is the larger of these possible paths; so, it’s max(h l, h l ) + 1, which is what our algorithm evaluates to. QED

Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Single-Elimination Tournaments –Binary Trees A Pattern for Induction Induction on Numbers Next Lecture Notes 37

An Induction Proof Pattern Type of Problem: Prove some property of a structure that is naturally defined in terms of itself. Part 1: Insight: how does the problem “break down” in terms of smaller pieces? Induction doesn’t help you with this part. It is not a technique to figure out patterns, only to prove them. Part 2: Proof. Establish that the property is true for your base case(s). Establish that it is true at each step of construction of a more complex structure. 38

A Pattern For Induction Theorem: 2 n teams can play in an n-round tournament. Proof: We proceed by induction on n. Base Case(s) (at most 2 0 teams can play in a 0-round tournament): Every tournament must have one unique winner. A zero-round tournament has no games; so, it can only include one (that is, 2 0 ) team: the winner. Inductive Step (for n greater than 0: if at most 2 n-1 teams can play in an (n-1)-round tournament, then at most 2 n teams can play in an n-round tournament): WLOG, let n be an integer greater than 0. Assume at most 2 n-1 teams can play in an (n-1)-round tournament. An n-round tournament is two (n-1)-round tournaments where the winners play each other (since there must be a single champion). By assumption, each of these may have at most 2 n-1 teams. So, the overall tournament has at most 2*2 n-1 = 2 n teams. This completes our induction proof. QED 39

A Pattern For Induction Theorem: P(n) is true for all n  some smallest number. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for a few simple cases): Prove with our other techniques. Which cases? Almost certainly the some smallest number. Maybe more. Make notes on which you need, as with a witness proof! Inductive Step (for n > the largest of your base cases: if P(. ) is true for whichever cases you need, then P(n) is true): WLOG, let n be greater than the largest of your base cases. Assume P(. ) is true for whichever cases you need. Break P(n) down in terms of the smaller case(s). The smaller cases are true, by assumption. Build back up to show that: P(n) is true. This completes our induction proof. QED 40

A Pattern For Induction  Which base cases? Almost certainly the smallest n. Otherwise, you don’t know yet. Do the rest of the proof now. Come back to the base case(s) when you know which one(s) you need! 41

A Pattern For Induction  Which values are we going to assume P(. ) is true for? Whichever we need. How do we know the ones we need? We don’t, yet. So… do the rest of the proof now. Come back to the assumption (“induction hypothesis”) when you know which one(s) you need! 42

A Pattern For Induction  What must n be larger than? The largest of your base cases. (Why? So you don’t assume the theorem true for something that’s too small, like a negative one round tournament.) But, you don’t know all your base cases yet. So…do the rest of the proof now. Come back to this bound once you know your base case(s). 43

A Pattern For Induction  How do we break the problem down in terms of smaller cases? THIS is the real core of every induction problem. Figure this out, and you’re ready to fill in the rest of the blanks! 44

A Pattern For Induction P(n) is ______________. Theorem: P(n) is true for all n  _______. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for _______): Prove each base case via your other techniques. Inductive Step (if P(. ) is true for ________________, then P(n) is true, for n > ____________): WLOG, let n be greater than ____________. Assume P(. ) is true for __________________. Break P(n) down in terms of the smaller case(s). The smaller cases are true, by assumption. Build back up to show that P(n) is true. This completes our induction proof. QED 45

Examples: Breaking down into a problem one smaller You want to prove P(n) for all n  1. You know that P(n) is true if P(n-1) is true. How do we fill in the blanks? 46 This is the most common style of insight, and the same as we had in our “# teams in a tournament” question. (It’s called “weak induction”.)

Examples: Breaking down into a problem one smaller You want to prove P(n) for all n  1. You know that P(n) is true if P(n-1) is true. How do we fill in the blanks? Theorem: P(n) is true for all n  _______. 47

Examples: Breaking down into a problem one smaller You want to prove P(n) for all n  1. You know that P(n) is true if P(n-1) is true. How do we fill in the blanks? Theorem: P(n) is true for all n  1. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for _______): Prove each base case via your other techniques. 48

Examples: Breaking down into a problem one smaller You want to prove P(n) for all n  1. You know that P(n) is true if P(n-1) is true. How do we fill in the blanks? Theorem: P(n) is true for all n  1. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for 1): Prove each base case via your other techniques. We only need n=1 because n=2 works based on n=1, and all subsequent cases also eventually break down into the n=1 case. Inductive Step (for n > _______, if P(. ) is true for ____________, then P(n) is true): 49

Examples: Breaking down into a problem one smaller You want to prove P(n) for all n  1. You know that P(n) is true if P(n-1) is true. How do we fill in the blanks? Theorem: P(n) is true for all n  1. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for 1): Prove each base case via your other techniques. Inductive Step (for n > 1: if P(. ) is true for n-1, then P(n) is true): WLOG, let n be greater than ____________. Assume P(. ) is true for __________________. 50

Examples: Breaking down into a problem one smaller You want to prove P(n) for all n  1. You know that P(n) is true if P(n-1) is true. How do we fill in the blanks? Theorem: P(n) is true for all n  1. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for 1): Prove each base case via your other techniques. Inductive Step (for n > 1: if P(. ) is true for n-1, then P(n) is true): WLOG, let n be greater than 1. Assume P(. ) is true for n-1. Break P(n) down in terms of the smaller case(s). The smaller cases are true, by assumption. Build back up to show that P(n) is true. This completes our induction proof. QED 51

Rephrased a bit (to get rid of P(. )) You want to prove P(n) for all n  1. You know that P(n) is true if P(n-1) is true. How do we fill in the blanks? Theorem: P(n) is true for all n  1. Proof: We proceed by induction on n. Base Case(s) (P(1) is true): Prove each base case via your other techniques. Inductive Step (for n > 1: if P(. ) is true for n-1, then P(n) is true): WLOG, let n be greater than 1. Assume P(n-1) is true. Break P(n) down in terms of the smaller case(s). The smaller cases are true, by assumption. Build back up to show that P(n) is true. This completes our induction proof. QED 52

Examples: Breaking down into all smaller problems You want to prove P(n) for all n  22. You know that P(n) is true if P(. ) is true for every integer from 22 up to n-1. How do we fill in the blanks? 53 This is the second most common style of insight, and the same as we had in our “prove the height algorithm correct” question. (It’s called “strong induction”; technically, we did “structural induction”.)

Examples: Breaking down into all smaller problems You want to prove P(n) for all n  22. You know that P(n) is true if P(. ) is true for every integer from 22 up to n-1. How do we fill in the blanks? Theorem: P(n) is true for all n  _______. 54

Examples: Breaking down into all smaller problems You want to prove P(n) for all n  22. You know that P(n) is true if P(. ) is true for every integer from 22 up to n-1. How do we fill in the blanks? Theorem: P(n) is true for all n  22. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for _______): Prove each base case via your other techniques. 55

Examples: Breaking down into all smaller problems You want to prove P(n) for all n  22. You know that P(n) is true if P(. ) is true for every integer from 22 up to n-1. How do we fill in the blanks? Theorem: P(n) is true for all n  22. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for 22): Prove each base case via your other techniques. For n=23, we just need n=22. For n=24, we just need n=22 and n=23… and n=23 breaks down in terms of n=22, and so on. Inductive Step (for n > _______, if P(. ) is true for ____________, then P(n) is true): 56

Examples: Breaking down into all smaller problems You want to prove P(n) for all n  22. You know that P(n) is true if P(. ) is true for every integer from 22 up to n-1. How do we fill in the blanks? Theorem: P(n) is true for all n  22. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for 22): Prove each base case via your other techniques. Inductive Step (for n > 22: if P(. ) is true for every integer from 22 up to n-1, then P(n) is true): WLOG, let n be greater than ____________. Assume P(. ) is true for __________________. 57

Examples: Breaking down into all smaller problems You want to prove P(n) for all n  22. You know that P(n) is true if P(. ) is true for every integer from 22 up to n-1. How do we fill in the blanks? Theorem: P(n) is true for all n  22. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for 22): Prove each base case via your other techniques. Inductive Step (for n > 22: if P(. ) is true for every integer from 22 up to n-1, then P(n) is true): WLOG, let n be greater than 22. Assume P(. ) is true for every integer from 22 up to n-1. Break P(n) down in terms of the smaller case(s). The smaller cases are true, by assumption. Build back up to show that P(n) is true. This completes our induction proof. QED 58

Rephrased a bit (to be more predicate logic-y) You want to prove P(n) for all n  22. You know that P(n) is true if P(. ) is true for every integer from 22 up to n-1. How do we fill in the blanks? Theorem: P(n) is true for all n  22. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for 22): Prove each base case via your other techniques. Inductive Step (for n > 22: if P(. ) is true for every integer from 22 up to n-1, then P(n) is true): WLOG, let n be greater than 22. Assume for all integers i where 22  i < n, P(i) is true. Break P(n) down in terms of the smaller case(s). The smaller cases are true, by assumption. Build back up to show that P(n) is true. This completes our induction proof. QED 59

Examples: breaking down into a problem half as big You want to prove P(n) for all n  7. You know that P(n) is true if P(  n/2  ) and P(  n/2  ) are both true (i.e., P(. ) is true for n/2 rounded down and n/2 rounded up). How do we fill in the blanks? 60 But, your insight may come in any form. Maybe you need problems half as large or one-third. Maybe you need problems that are 7 smaller. Maybe you need the problems that are 1, 2, and 3 smaller. Regardless, the pattern is the same!

Examples: breaking down into a problem half as big You want to prove P(n) for all n  7. You know that P(n) is true if P(  n/2  ) and P(  n/2  ) are both true (i.e., P(. ) is true for n/2 rounded down and n/2 rounded up). How do we fill in the blanks? Theorem: P(n) is true for all n  _______. 61

Examples: breaking down into a problem half as big You want to prove P(n) for all n  7. You know that P(n) is true if P(  n/2  ) and P(  n/2  ) are both true (i.e., P(. ) is true for n/2 rounded down and n/2 rounded up). How do we fill in the blanks? Theorem: P(n) is true for all n  7. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for _______): Prove each base case via your other techniques. 62

Examples: breaking down into a problem half as big You want to prove P(n) for all n  7. You know that P(n) is true if P(  n/2  ) and P(  n/2  ) are both true (i.e., P(. ) is true for n/2 rounded down and n/2 rounded up). How do we fill in the blanks? Theorem: P(n) is true for all n  7. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for n = 7, 8, 9, 10, 11, 12, 13): Prove each base case via your other techniques. (We need all the way up to 13 because only at 14/2 do we reach a base case. From 15 on, we always eventually hit a base case.) Inductive Step (for n > _______, if P(. ) is true for ____________, then P(n) is true): 63

Examples: breaking down into a problem half as big You want to prove P(n) for all n  7. You know that P(n) is true if P(  n/2  ) and P(  n/2  ) are both true (i.e., P(. ) is true for n/2 rounded down and n/2 rounded up). How do we fill in the blanks? Theorem: P(n) is true for all n  7. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for n = 7, 8, 9, 10, 11, 12, 13): Prove each base case via your other techniques. Inductive Step (for n > 13: if P(. ) is true for  n/2  and  n/2 , then P(n) is true): WLOG, let n be greater than ____________. Assume P(. ) is true for __________________. 64

Examples: breaking down into a problem half as big You want to prove P(n) for all n  7. You know that P(n) is true if P(  n/2  ) and P(  n/2  ) are both true (i.e., P(. ) is true for n/2 rounded down and n/2 rounded up). How do we fill in the blanks? Theorem: P(n) is true for all n  7. Proof: We proceed by induction on n. Base Case(s) (P(. ) is true for n = 7, 8, 9, 10, 11, 12, 13): Prove each base case via your other techniques. Inductive Step (for n > 13: if P(. ) is true for  n/2  and  n/2 , then P(n) is true): WLOG, let n be greater than 13. Assume P(. ) is true for  n/2  and  n/2 . Break P(n) down in terms of the smaller case(s). The smaller cases are true, by assumption. Build back up to show that P(n) is true. This completes our induction proof. QED 65

Rephrased slightly (to get rid of P(. )) You want to prove P(n) for all n  7. You know that P(n) is true if P(  n/2  ) and P(  n/2  ) are both true (i.e., P(. ) is true for n/2 rounded down and n/2 rounded up). How do we fill in the blanks? Theorem: P(n) is true for all n  7. Proof: We proceed by induction on n. Base Case(s) (P(7), P(8), P(9), P(10), P(11), P(12), P(13)): Prove each base case via your other techniques. Inductive Step (for n > 13: if P(. ) is true for  n/2  and  n/2 , then P(n) is true): WLOG, let n be greater than 13. Assume P(  n/2  ) and P(  n/2  ) are both true. Break P(n) down in terms of the smaller case(s). The smaller cases are true, by assumption. Build back up to show that P(n) is true. This completes our induction proof. QED 66

Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Single-Elimination Tournaments –Binary Trees A Pattern for Induction Induction on Numbers Next Lecture Notes 67

Natural Numbers and Induction Can we prove things inductively about the natural numbers without a recursive data definition? Are they “self-referential”? Here’s one answer: A natural number is: -The next number after (successor of) a natural number 68

Natural Numbers and Induction Can we prove things inductively about the natural numbers without a recursive data definition? Are they “self-referential”? But, let’s just try one! Problem: What is the sum of the natural numbers 0..n? 69

Partly Worked Problem: Sum of 0..n Partly Worked Problem: What is the sum of the natural numbers 0..n? Induction doesn’t help with insight... But spoilers do: n(n+1)/2. Now, how do we prove it? 70

Partly Worked Problem: Sum of 0..n Partly Worked Problem: Prove that the sum of the natural numbers 0..n is n(n+1)/2. Is this self-referential? If we knew the sum of the numbers 0..(n-1), would it tell us anything about the sum of the numbers 0..n? 71

Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Single-Elimination Tournaments –Binary Trees A Pattern for Induction Induction on Numbers Next Lecture Notes 72

Learning Goals: In-Class By the end of this unit, you should be able to: –Establish properties of self-referential structures using inductive proofs that naturally build on those self-references. Note: this learning goal is not yet looking for formal inductive proofs. Instead, we’re just exploring how we work with things that are defined in terms of themselves. 73

Next Lecture Learning Goals: Pre-Class By the start of class, you should be able to: –Given a theorem to prove and the insight into how to break the problem down in terms of smaller problems, write out the skeleton of an inductive proof including: the base case(s) that need to be proven, the induction hypothesis, and the inductive step that needs to be proven. So, take what we did in this lecture and use the textbook readings to formalize your understanding. We will have much more practice on inductive proofs. 74

Next Lecture Prerequisites See the Mathematical Induction Textbook Sections at the bottom of the “Textbook and References” page.Mathematical Induction Textbook Sections 75