CSC236H Lecture 1

Sadia Sharmin

September 10, 2018

Outline

Course Introduction

Mathematical Induction

Welcome to CSC236

I CSC236 is your first CS-theory course

Welcome to CSC236

I This course is much more “CS” than CSC108 (and maybeeven CSC148)

I We’re going to studyI Program correctnessI Program e�ciencyI How to design e�cient algorithms

Welcome to CSC236

I What you learn in this course is going to be very useful!

Evaluation

I Pre-lecture reading quizzes (5%)

I Class participation (5%)

I Two tests (better one worth 18%, other worth 12%)

I Final exam (45%)

I Five problem sets (15%)

Huh? I Can’t Sleep in Class?

Huh? I Can’t Sleep in Class?

Please don’t.

I Class participation is worth 5%

I But that’s not the most important reason

I Most important reason: learning is enhanced when we areactively engaged in and questioning the material

I You will have opportunities to solve and discuss solutions toproblems rather than watching me solve problems for you

How Lectures Work

I We are using a technique called Peer Instruction (PI)

I I present a short summary of a key topic and then ask aquestion

I You think about it for a minute or two, on your own (notalking just yet, please)

I You anonymously report your answers using your clicker

I Then, you discuss that same question with a small group ofyour neighbors, trying to convince them of your reasoning andreach consensus

I You report your revised answers, and we discuss as a class

How Lectures Work...

I On occasion we will extend this patternI e.g. I might ask you a follow-up question but with no group

discussionI e.g. I might make up new questions on the spot if I notice

di�culties

I I’ll clearly indicate what we are doing for each question

What’s the Point?

I Helps you assess your own understandingI Did you answer correctly? Why did you choose your answer?

Why are the other answers incorrect?

I Helps you think through the ideas of other studentsI Explanations of other students (correct or incorrect) can help

you gain a better understanding of the topic

I Lets me know where di�culties are arising, and how toaddress them

Where’s the Evidence?I A team of researchers, has researched PI in CS since 2010I That’s eight years of researchI We have studied student learning, retention of knowledge,

attitudes, comparisons to traditional lecture-based courses,benefits of individual thinking, benefits of group discussion . . .you name it

I PI simply works better than traditional lecture

How we will use Clickers

I Please purchase an i>clicker remote for this course (can buyfrom bookstore)

I Please register it on Quercus. How-to available on course website.

I I’ll ask questions on the screen during lecture, to which youcan respond with your clicker

I After discussion, I’ll display a graph with the class results onthe screen

I We’ll discuss the questions and answers

I You get points for participating

How to Vote

I Make sure that the clicker is powered on

I A blue “Power” light will appear at the top of the remote

I When I ask a question in class (and start the timer), select A,B, C, D, or E as your vote

I Green light: your vote was sent AND received

I Red light: error! Vote again!

I You can always change your vote while the timer is going —your last vote is the one that counts

Clicker Tips

I If you bought a used clicker, replace all AAA batteries

I Don’t use Duracell as they are TOO short for the casing

I Don’t use rechargeable batteries. They harm the clicker

I Register your clicker before our next class

I Before using a new clicker for the first time, pull the plastictab out of the battery compartment

I Bring your clicker to every class. Make sure it is on beforevoting!

Sample Clicker Question

What makes you the happiest?

I A. Cat videos

I B. Free food

I C. Having 9am class with Sadia

I D. Board games

I E. Expanding my knowledge about the theory of computation,e.g. learning how to prove my algorithms are correct, learninghow to calculate the runtime e�ciency of recursivealgorithms, and other such amazing stu↵

Reading Quizzes

I Before each lecture, you’ll complete a short reading quiz onlineI Why?

I You have to be prepared for class for our discussions to beproductive

I I can then focus on di�cult parts of the material and newexamples

I I will not teach you the course notes in class (you know how toread!)

Participation and Reading Quiz Grading

I You are graded on the quality of your participation in class,not on the correctness of your answers

I You are graded on reasonable completion of readingquizzes, not on the correctness of your answers

I In this course, we use an absolute grading scale (nobell-curve)

I I want you to feel comfortable discussing and understandingideas with other students

Problem Sets

I Problem sets are due approximately every two weeks

I You can work individually or in a group of at most 3 students

I Problem sets contain an unmarked “preliminary” exercise.Please do that early!

Submission Policy

I LaTeX is a system for typesetting documents

I Many computer scientists, mathematicians, physicists, etc.use LaTeX

I I am requiring you to submit all problem sets using LaTeXI You must submit both your .tex source and resulting .pdf

file

I Please check the course website for useful introductoryresources

Academic Integrity - when you are bad vs. good

Academic Integrity tips

I You all know what I expect by now

I Do not discuss problem sets outside of your group

I Please don’t “help” others by helping them solve the problemsets!

I Please don’t put others in awkward positions by asking themfor help on a problem set

I What you can do to helpI Discuss examples from lecture and the course materialsI Discuss “preliminary” questions

Tutorials

I Tutorial attendance is not required for marks

I But it is a great way to practice course material with thesupport of a TA and other students!

I TAs can help you understand course material and workthrough examples and previous problem sets

I Feel free to bring questions!

Course Materials

ICSC236 Lecture Notes is our core text for the course,available for free from the course web site. We will follow itclosely

I Additional readings as indicated on the course web site.

Course Decisions

Course Decisions

We will refer back to these! In this course:

I If not specified, our proofs are on the natural numbers0, 1, . . . , n

I log n means “log base 2 of n”. The default base is 2!

I Use of limits for proving bounds is not allowed

I Regular expressions must use only syntax introduced in thiscourse

Outline

Course Introduction

Mathematical Induction

ConcepTest

Which of the following is not a predicate?

I A. P1(n) :Pn

i=0 i =n(n+1)

2

I B. P2(n) : n is prime or is equal to the product of two or moreprimes

I C. P3(n) : n is a perfect square

I D. These are all predicates

Mathematical Induction

I Mathematical induction is a proof technique that can be usedto prove that a certain statement is true for all naturalnumbers (or all natural numbers starting from number b)

I Symbolically: it lets us prove claims of the form 8n 2 N,P(n)

I We’ll start with Simple (or Weak) induction . . .

Let’s visualize it.

IP(n) : “n-th domino falls backwards”

I Note P(n) makes sense for n > 0.

I We are given “P(1) falls’ backwards” is true.

I If P(K ) falls, then P(k + 1) will fall.

I We conclude all dominos fall backwards.

ConcepTest

We want to prove that P(n) is true for all integers n � b. Which isnot a valid proof?

I A. (1) Prove P(b); (2) prove, for all n � b, P(n) ) P(n + 1)

I B. (1) Prove P(b); (2) prove, for all n > b, P(n � 1) ) P(n)

I C. (1) Prove P(b); (2) prove, for all n � b + 3,P(n) ) P(n + 1)

I D. (1) Prove P(b); (2) prove, for all n � b + 3,P(n � 3) ) P(n � 2)

I E. More than one of the above is invalid

Base Case and Inductive Step

Most often, we use the P(n) ) P(n + 1) form of induction.For some predicate P :

I (1) When we show that P(b) is true for some b, we areproving the base case

I (2) When we show that P(n) ) P(n + 1) for all n � b, weare proving the inductive case

I To help us prove the inductive case, we assume and use theInductive Hypothesis (I.H.) that P(n) is true

Mathematical Induction Proof Recipe

We are given a problem that depends on a natural number, wantto use proof by induction. Here is what we need to do:

1. Identify and write down the predicate that depends on anatural number. Let b be the smallest number the predicate isdefined.

2. Prove the base case P(b).

3. Let n be a generic natural number � b.

4. Assume P(n) holds. Derive P(n + 1).

5. Conclude the property speicifed by the given problem holds.

Q: If at least one the steps (above) fails, can we conclude thespecified property does not hold?

Exercise 1

Exercise 1 on handout.

ConcepTest

Read Presentation 1 on the handout. Which of the following istrue?

I A. The proof is valid

I B. The proof is valid, but the claim is false

I C. The proof is invalid; line (a) is incorrect

I D. The proof is invalid; line (b) is incorrect