Transcript Introduction to Induction

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CPSC 121: Models of Computation
2008/9 Winter Term 2
Introduction to Induction
Steve Wolfman
Lecture Prerequisites
Read Section 4.1.
Note: 4.1 is background information. We don’t
intend to directly assess this material, but
assignment and exam questions may require
skills and knowledge from 4.1 to answer.
Solve problems like Exercise Set 4.1, #1-8, 1214, 18-31, 37-44, 52-60.
Complete the open-book, untimed quiz on Vista
that’s due before the next class.
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.
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.
Problem: Single-Elimination
Tournaments
In a single-elimination tournament, during
each round pairs of teams play each other,
with the losing team eliminated from the
tournament. The tournament ends when
only one team remains.
Problem 1: Determine the maximum
number of teams in an n-round singleelimination tournament.
Problem 2: Prove your result.
Problem: Number of Nodes
in a Binary Tree
A “binary tree” is a root “node” and a collection
of “child nodes” beneath the root.
Each node is an object that has up to two
children—a left child and a right child—each
of which are themselves binary trees.
A binary tree’s “height” is the maximum number
of steps it takes to get from the root down to
a node with no children.
Problem 1: What’s the maximum number of
nodes in a binary tree of height n?
Problem 2: Prove your result.
Problem: 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.
An Induction Proof Pattern
Type of Problem: Prove some property of a
structure that is naturally defined in terms of
itself.
Part 1: Insight. 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 holds
for simple structures. Establish that it holds
at each step of construction of a more
complex structure.
Natural Numbers and Induction
Can we prove things about the natural
numbers using induction? Are they “selfreferential”?
Let’s try one...
Problem: What is the sum of the natural
numbers 0..n?
Partly Worked Problem:
Sum of 0..n
Partly Worked Problem: What is the sum
of the natural numbers 0..n?
Insight: It’s n(n+1)/2.
Now, how do we prove it?
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?
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.
Next Lecture Learning Goals:
Pre-Class
By the start of class, you should be able to:
– Given a theorem to prove stated in terms of its
induction variable (i.e., usually, in terms of n),
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.
In other words, 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.
Next Lecture Prerequisites
Read Section 4.2-4.4.
Solve (or at least “set up”, by writing the base
case(s) that need to be proved, the induction
hypothesis, and the inductive step that needs
to be proved) problems like Exercise Set 4.2
#1-2, 5-17, and 31; 4.3 #1, 8-27, 29, and 3031, and 4.4 #1-7, 9-10, and 24.
Solve problems like Exercise Set 4.2 #3-4 and
32, 4.3 #6-7 and 30-31, and 4.4 #17.
Complete the open-book, untimed quiz on
WebCT that’s due before the next class.