Pigeonhole Principle

Transcription

1 Pigeonhole Principle TUT0003 CSC/MATA67 October 19th, 2017

2 Housekeeping Quiz 3 handed back Ex4 marking + Ex3 marks A1 is out! CSEC-S meeting tomorrow (3-5pm in IC200) CSEC Chess AI Seminar (6-8pm, IC230) Midterm reminder *Insert eerie music here*

3 Quiz today? But is there a quiz today?

4 Quiz today? But is there a quiz today? Of course there is, get bamboozled.

5 Clearing up Confusion on Contradiction... Contradiction: Defn: (Noun) A combination of statements, ideas, or features of a situation that are opposed to one another. How does the contradiction of a statement being false mean that the original statement holds true? The big reason to use a proof by contradiction? The original proof is too hard to prove.

6 Clearing up Confusion on Contradiction... Example: n Z, n is odd n is even. Suppose both n and n are odd. By the definition of odd numbers, there exists integers k and j such that n = 2k + 1 and n = 2j + 1 Substitute for n, giving us: 2k + 1 = n = (2j + 1) = 8j 3 + 3(2j) 2 + 3(2j) + 6 2k = 8j j 2 + 6j + 5 Divide by 2 and rearrange: k 4j 3 6j 2 3j = 5 2 This doesn t hold under integer subtraction, thus breaking our contradiction, proving the original statement.

7 What to do for proof by contradiction: case by case Statement P(x) P(x) Q(x) How to Contradict it (Assume) P(x) P(x) Q(x)

8 Pigeonhole Principle Def n : If k + 1 or more pigeons are distributed among k pigeonholes, then at least one pigeonhole contains two or more pigeons. Layman s terms: If there s more items to store than there are individual boxes, someone s gotta share.

9 Pigeonhole Principle: A step-by-step process Applying the Pigeonhole Principle is done using the following process: 1 Figure out what your pigeons/items are

10 Pigeonhole Principle: A step-by-step process Applying the Pigeonhole Principle is done using the following process: 1 Figure out what your pigeons/items are 2 Figure out what your pigeonholes/slots are, ensure there are less pigeonholes/slots than there are pigeons/items

11 Pigeonhole Principle: A step-by-step process Applying the Pigeonhole Principle is done using the following process: 1 Figure out what your pigeons/items are 2 Figure out what your pigeonholes/slots are, ensure there are less pigeonholes/slots than there are pigeons/items 3 Assign a rule to the pigeonholes. What happens once two items share a slot?

12 Pigeonhole Principle: A step-by-step process Applying the Pigeonhole Principle is done using the following process: 1 Figure out what your pigeons/items are 2 Figure out what your pigeonholes/slots are, ensure there are less pigeonholes/slots than there are pigeons/items 3 Assign a rule to the pigeonholes. What happens once two items share a slot? 4 Apply it to your overarching proof

13 Pigeonhole Principle: A step-by-step process Applying the Pigeonhole Principle is done using the following process: 1 Figure out what your pigeons/items are 2 Figure out what your pigeonholes/slots are, ensure there are less pigeonholes/slots than there are pigeons/items 3 Assign a rule to the pigeonholes. What happens once two items share a slot? 4 Apply it to your overarching proof 5 QED???

14 Pigeonhole Principle: Example Q: Prove that if seven distinct numbers are selected from {1, 2,..., 11}, then some two of these numbers sum to 12. What are our pigeons? What are our pigeonholes?

15 Pigeonhole Principle: Example Q: Prove that if seven distinct numbers are selected from {1, 2,..., 11}, then some two of these numbers sum to 12. What are our pigeons? What are our pigeonholes? Pigeons: The numbers in our set. Pigeonholes: Tuples containing numbers whose sum is 12. {1, 11}, {2, 10}, {3, 9}, {4, 8}, {5, 7}, {6}

16 Pigeonhole Principle: Example Q: Prove that if five points are selected from the interior of a 1x1 square, then there are two points whose distance is less than 2 2.

17 Pigeonhole Principle: Example Q: Prove that if five points are selected from the interior of a 1x1 square, then there are two points whose distance is less than 2 2. Pigeons: The five points within the square. Pigeonholes: Four 0.5x0.5 Sub-spaces created by joining the midpoints of opposite edges of the square.

18 Pigeonhole Principle: Example Q: Prove that within in a group of n people, there exists a pair of people who have the same number of friends. NOTE: Everyone is friends with at least someone who isn t themselves.

19 Pigeonhole Principle: Example Q: Prove that within in a group of n people, there exists a pair of people who have the same number of friends. NOTE: Everyone is friends with at least someone who isn t themselves. Pigeons: The number of people in the group. Pigeonholes: The number of friends a person has.