![Page 1: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/1.jpg)
Lecture 3.1: Mathematical Induction
CS 250, Discrete Structures, Fall 2014
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag
![Page 2: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/2.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Course Admin Mid-Term 1
Hope it went well! Thanks for your cooperation running it
smoothly We are grading them now, and should have
the results no less than two weeks Solution will be provided soon Questions?
![Page 3: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/3.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Course Admin HW2 (don’t forget)
Due Oct 14 (Tues)
![Page 4: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/4.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Outline
Mathematical Induction Principle Examples Why it all works
![Page 5: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/5.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionSuppose we have a sequence of propositions
which we would like to prove:P (0), P (1), P (2), P (3), P (4), … P (n), …EG: P (n) = “The sum of the first n positive odd numbers is
equal to n2”We can picture each proposition as a domino:
P (n)
![Page 6: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/6.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionSo sequence of propositions is a sequence
of dominos.
…
P (n+1)P (n)P (2)P (1)P (0)
…
![Page 7: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/7.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionWhen the domino falls, the corresponding
proposition is considered true:
P (n)
![Page 8: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/8.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionWhen the domino falls (to right), the
corresponding proposition is considered true:
P (n)true
![Page 9: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/9.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionSuppose that the dominos satisfy two
constraints.1) Well-positioned: If any domino falls (to
right), next domino (to right) must fall also.
2) First domino has fallen to right
P (0)true
P (n+1)P (n)
![Page 10: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/10.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionSuppose that the dominos satisfy two
constraints.1) Well-positioned: If any domino falls to
right, the next domino to right must fall also.
2) First domino has fallen to right
P (0)true
P (n+1)P (n)
![Page 11: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/11.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionSuppose that the dominos satisfy two
constraints.1) Well-positioned: If any domino falls to
right, the next domino to right must fall also.
2) First domino has fallen to right
P (0)true
P (n)true
P (n+1)true
![Page 12: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/12.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionThen can conclude that all the dominos
fall!
…
P (n+1)P (n)P (2)P (1)P (0)
![Page 13: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/13.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionThen can conclude that all the dominos
fall!
…
P (n+1)P (n)P (2)P (1)P (0)
![Page 14: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/14.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionThen can conclude that all the dominos
fall!
…P (0)true
P (n+1)P (n)P (2)P (1)
![Page 15: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/15.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionThen can conclude that all the dominos
fall!
…P (0)true
P (1)true
P (n+1)P (n)P (2)
![Page 16: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/16.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionThen can conclude that all the dominos
fall!
P (2)true
…P (0)true
P (1)true
P (n+1)P (n)
![Page 17: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/17.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionThen can conclude that all the dominos
fall!
P (2)true
…P (0)true
P (1)true
P (n+1)P (n)
![Page 18: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/18.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionThen can conclude that all the dominos
fall!
P (2)true
…P (0)true
P (1)true
P (n)true
P (n+1)
![Page 19: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/19.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionThen can conclude that all the dominos
fall!
P (2)true
…P (0)true
P (1)true
P (n)true
P (n+1)true
![Page 20: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/20.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical InductionPrinciple of Mathematical Induction: If:1) [basis] P (0) is true2) [induction] k P(k)P(k+1) is true
Then: n P(n) is trueThis formalizes what occurred to dominos.
P (2)true
…P (0)true
P (1)true
P (n)true
P (n+1)true
![Page 21: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/21.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Exercise 1Use induction to prove that the sum of the first
n odd integers is n2.Prove a base case (n=1)
Base case (n=1): the sum of the first 1 odd integer is 12. Yup, 1 = 12.
Prove P(k)P(k+1)
Assume P(k): the sum of the first k odd ints is k2. 1 + 3 + … + (2k - 1) = k2
Prove that 1 + 3 + … + (2k - 1) + (2k + 1) = (k+1)2
1 + 3 + … + (2k-1) + (2k+1) =
k2 + (2k + 1)= (k+1)2 By arithmetic
![Page 22: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/22.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Exercise 2
Prove that 11! + 22! + … + nn! = (n+1)! - 1, nBase case (n=1): 11! = (1+1)! - 1?Yup, 11! = 1, 2! - 1 = 1
Assume P(k): 11! + 22! + … + kk! = (k+1)! - 1Prove that 11! + … + kk! + (k+1)(k+1)! = (k+2)! - 111! + … + kk! + (k+1)
(k+1)! = (k+1)! - 1 + (k+1)
(k+1)!= (1 + (k+1))(k+1)! - 1= (k+2)(k+1)! - 1
= (k+2)! - 1
![Page 23: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/23.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Exercises 3 and 4 (have seen before?)
1. Recall sum of arithmetic sequence:
2. Recall sum of geometric sequence:
2
)1(
1
nni
n
i
1
)1(...
12
0
r
raarararaar
nn
n
i
i
![Page 24: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/24.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical Induction - why does it work?
Proof of Mathematical Induction:
We prove that (P(0) (k P(k) P(k+1))) (n P(n))
Proof by contradiction.
Assume1. P(0)2. k P(k) P(k+1)3. n P(n) n P(n)
![Page 25: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/25.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Mathematical Induction - why does it work?
Assume1. P(0)2. k P(k) P(k+1)3. n P(n) n P(n)Let S = { n :
P(n) }
Since N is well ordered, S has a least element. Call it k.
What do we know? P(k) is false because it’s in S. k 0 because P(0) is true. P(k-1) is true because P(k) is the least
element in S.
But by (2), P(k-1) P(k). Contradicts P(k-1) true, P(k)
false.
Done.
![Page 26: Lecture 3.1: Mathematical Induction CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag](https://reader035.vdocuments.us/reader035/viewer/2022070409/56649e765503460f94b77622/html5/thumbnails/26.jpg)
04/21/23Lecture 3.1 -- Mathematical
Induction
Today’s Reading Rosen 5.1