Pigeonhole Principle – Page 1CPTR311 – Discrete Structures CPTR311 Discrete Structures Pigeonhole Principle Reading: Kolman, Section 3.3.

Slides:



Advertisements
Similar presentations
The Pigeonhole Principle:
Advertisements

Mega Math Mania An Interactive Videoconference Competition for Math Problem Solving.
The Pigeon hole problem
MAT 2720 Discrete Mathematics
This trick relies on three nice ideas from discrete mathematics: pigeonhole principle 5 pigeons (cards) in 4 holes (suits) means at least two pigeons are.
The Pigeonhole Principle College of Information Technology & Design
The Pigeonhole Principle
Practice Quiz 3 Recursive Definitions Relations Basic Counting Pigeonhole Principle Permutations & Combinations Discrete Probability.
Breakfast Bytes: Pigeons, Holes, Bridges and Computers Alan Kaylor Cline November 24, 2009.
CISC1400: Discrete Structure Introduction 1 Dr. Zhang, Fall 2011, Fordham Univ.
COUNTING AND PROBABILITY
Application: The Pigeonhole Principle Lecture 37 Section 7.3 Wed, Apr 4, 2007.
Recursively Defined Functions
The Pigeonhole Principle
Quiz highlights Probability of the song coming up after one press: 1/N. Two times? Gets difficult. The first or second? Or both? USE THE MAIN HEURISTICS:
PIGEONHOLE PRINCIPLE. Socks You have a drawer full of black and white socks. Without looking in the drawer, how many socks must you pull out to be sure.
The Pigeonhole Principle 6/11/ The Pigeonhole Principle In words: –If n pigeons are in fewer than n pigeonholes, some pigeonhole must contain at.
Discrete Structures Chapter 5 Pigeonhole Principle Nurul Amelina Nasharuddin Multimedia Department.
The Pigeonhole Principle
Discrete Mathematics Lecture 7 Alexander Bukharovich New York University.
CSE115/ENGR160 Discrete Mathematics 04/10/12
Stat 245 Recitation 11 10/25/2007 EA :30am TA: Dongmei Li.
Pigeonhole.
Discrete Mathematics Lecture 7 Harper Langston New York University.
1 The Pigeonhole Principle CS/APMA 202 Rosen section 4.2 Aaron Bloomfield.
Fall 2015 COMP 2300 Discrete Structures for Computation
The Pigeonhole (Dirichlet’s box) Principle
Counting. Product Rule Example Sum Rule Pigeonhole principle If there are more pigeons than pigeonholes, then there must be at least one pigeonhole.
Fall 2002CMSC Discrete Structures1 One, two, three, we’re… Counting.
September1999 CMSC 203 / 0201 Fall 2002 Week #8 – 14/16 October 2002 Prof. Marie desJardins.
1 CSC 321: Data Structures Fall 2013 Counting and problem solving  mappings, bijection rule  sequences, product rule, sum rule  generalized product.
The Pigeonhole Principle. The pigeonhole principle Suppose a flock of pigeons fly into a set of pigeonholes to roost If there are more pigeons than pigeonholes,
5.2 The Pigeonhole Principle
Discrete Structures Counting (Ch. 6)
Unit II Discrete Structures Relations and Functions SE (Comp.Engg.)
1 Melikyan/DM/Fall09 Discrete Mathematics Ch. 7 Functions Instructor: Hayk Melikyan Today we will review sections 7.3, 7.4 and 7.5.
1 Chapter 2 Pigeonhole Principle. 2 Summary Pigeonhole principle –simple form Pigeonhole principle –strong form Ramsey’s theorem.
CS Lecture 26 Monochrome Despite Himself. Pigeonhole Principle: If we put n+1 pigeons into n holes, some hole must receive at least 2 pigeons.
1 The Pigeonhole Principle CS 202 Epp section 7.3.
2/24/20161 One, two, three, we’re… Counting. 2/24/20162 Basic Counting Principles Counting problems are of the following kind: “How many different 8-letter.
Introduction Suppose that a password on a computer system consists of 6, 7, or 8 characters. Each of these characters must be a digit or a letter of the.
1 The Pigeonhole Principle CS 202 Epp section ??? Aaron Bloomfield.
The Pigeonhole Principle. Pigeonhole principle The pigeonhole principle : If k is a positive integer and k+1 or more objects are placed into k boxes,
Pigeonhole Principle. If n pigeons fly into m pigeonholes and n > m, then at least one hole must contain two or more pigeons A function from one finite.
Discrete Mathematics. Exercises Exercise 1:  There are 18 Computer Science (CS) majors and 325 Business Administration (BA) majors at a college.
Main Menu Main Menu (Click on the topics below) Pigeonhole Principle Example Generalized Pigeonhole Principle Example Proof of Pigeonhole Principle Click.
ICS 253: Discrete Structures I Counting and Applications King Fahd University of Petroleum & Minerals Information & Computer Science Department.
Section The Product Rule  Example: How many different license plates can be made if each plate contains a sequence of three uppercase English letters.
COUNTING Discrete Math Team KS MATEMATIKA DISKRIT (DISCRETE MATHEMATICS ) 1.
The Pigeonhole Principle
CSC 321: Data Structures Fall 2015 Counting and problem solving
PIGEONHOLE PRINCIPLE.
The Pigeonhole Principle
Discrete Structures for Computer Science
COCS DISCRETE STRUCTURES
CSC 321: Data Structures Fall 2016 Counting and proofs
The Pigeonhole Principle
PIGEONHOLE PRINCIPLE.
PIGEONHOLE PRINCIPLE.
Xuan Guo Lab 9 Xuan Guo
The Pigeonhole Principle
PIGEONHOLE PRINCIPLE.
The Pigeonhole (Dirichlet’s box) Principle
CSC 321: Data Structures Fall 2018 Counting and proofs
Counting techniques Basic Counting Principles, Pigeonhole Principle, Permutations and Combinations.
The Pigeonhole Principle
The Pigeon hole problem
Representations of Integers
Probability Chances Are… You can do it! Activity #2.
Miniconference on the Mathematics of Computation
Presentation transcript:

Pigeonhole Principle – Page 1CPTR311 – Discrete Structures CPTR311 Discrete Structures Pigeonhole Principle Reading: Kolman, Section 3.3

Pigeonhole Principle – Page 2CPTR311 – Discrete Structures The Pigeonhole Principle Suppose a flock of pigeons fly into a set of pigeonholes to roost If there are more pigeons than pigeonholes, then there must be at least 1 pigeonhole that has more than one pigeon in it If there are n+1 pigeons, which must fit into n pigeonholes then some pigeonhole contains 2 or more pigeons

Pigeonhole Principle – Page 3CPTR311 – Discrete Structures Generalized Pigeonhole Principle If N objects are placed into k boxes, there is at least one box containing  N/k  objects

Pigeonhole Principle – Page 4CPTR311 – Discrete Structures Key to solving any problem Identify two things: –Number of pigeons in the problem  N –Number of holes  k –Then at least  N/k  will have same property.

Pigeonhole Principle – Page 5CPTR311 – Discrete Structures Examples Assume you have a drawer containing a random distribution of a dozen brown socks and a dozen black socks. It is dark, so how many socks do you have to pick to be sure that among them there is a matching pair? –Number of pigeon (number of socks to pick)  N (unknown) –Number of holes (colors)  2 –At least two socks should have the same color  N/2  = 2  N = 3

Pigeonhole Principle – Page 6CPTR311 – Discrete Structures Examples How many numbers must be selected from set {1, 2, 3, 4, 5, 6} to guarantee that at least one pair add to 7? –Solved in two steps: how many pairs add to 7 {1, 6}, {2, 5}, {3, 4}  3 sets –Number of pigeons (number of numbers to pick)  N (unknown) –Number of holes (sets)  3 –At least two numbers fall into one set  N/3  = 2  N = 4

Pigeonhole Principle – Page 7CPTR311 – Discrete Structures Examples Among 100 people, there are at least  100/12  = 9 born on the same month How many students in a class must there be to ensure that 6 students get the same grade (one of A, B, C, D, or F)? –The “boxes” are the grades. Thus, k = 5 –Thus, we set  N/5  = 6 –Lowest possible value for N is 26

Pigeonhole Principle – Page 8CPTR311 – Discrete Structures Examples 6 computers on a network are connected to at least 1 other computer. Show there are at least two computers that are have the same number of connections The number of boxes, k, is the number of computer connections –This can be 1, 2, 3, 4, or 5 The number of pigeons, N, is the number of computers –That’s 6 By the generalized pigeonhole principle, at least one box must have  N/k  objects –  6/5  = 2 –In other words, at least two computers must have the same number of connections