CSE 20 – Discrete Mathematics - PowerPoint Presentation

Slide1

CSE 20 – Discrete Mathematics

Dr. Cynthia Bailey LeeDr. Shachar Lovett

Peer Instruction in Discrete Mathematics by 

Cynthia

Lee

is

licensed under a 

Creative Commons Attribution-

NonCommercial

-

ShareAlike

4.0 International License

.

Based on a work at 

http://peerinstruction4cs.org

.

Permissions beyond the scope of this license may be available at 

http://peerinstruction4cs.org

.

Slide2

Today’s Topics:

RelationsEquivalence relations

2Slide3

1. Relations

3Slide4

Relations are graphs

Think of relations as graphsxRy means “there in an edge xy

”Is R ? R ?BothNeither4Slide5

Relations are graphs

What does this relation capture? xRy meansx>y

x=yx divides yx+yNone/more than one5234

15

6Slide6

Types of relations

A relation is symmetric if xRyyRx.

That is, if the graph is undirectedWhich of the following is symmetric x<yx divides yx and y have the same signxyNone/more than one6Slide7

Types of relations

A relation is reflexive if xRx

is true for all xThat is, the graph has loops in all verticesWhich of the following is reflexivex<yx divides yx and y have the same signxyNone/more than one7Slide8

Types of relations

A relation is transitive if xRy

 yRz  xRzThis is less intuitive… will show equivalent criteria soonWhich of the following is transitivex<yx divides yx and y have the same signxyNone/more than one8Slide9

Types of relations

A relation is transitive if xRy

 yRz  xRzTheorem: Let G be the graph corresponding to a relation R. R is transitive iff whenever you can reach from x to y in G then the edge xy is in G.Try to prove yourself first9Slide10

Types of relations

Theorem: R is transitive iff when you can reach from x to y in G then the edge xy

is also in G.Proof (sufficient): Assume the graph G has this property. We will show R is transitive. Let x,y,z be such that xRy and yRz hold. In the graph G we can reach from x to z via the path xyz. So by assumption on G xz is also an edge in G. Hence xRz so R is transitive. 10Slide11

Types of relations

Theorem: R is transitive iff when you can reach from x to y in G then the edge

xy is also in G.Proof (necessary) by contradiction:Assume by contradiction R is transitive but G doesn’t have this property. So, there are vertices x,y with a path xv1…vky in G but where there is no edge xy. Choose such a pair x,y with minimal path length k. We divide the proof to cases.Case 1: k=0. So xy in G. Contradiction. Case 2: k=1. Since R is transitive then xv1 and v1y imply xy

. Contradiction.Case 3: k>1. Then xvk must be in G since the path has length k-1 and we assumed the path from x to y is of minimal length. So in fact

xv

k

y

. Contradiction.

QED

11Slide12

2. Equivalence relations

12Slide13

Equivalence relations

13Definition: a relation is an

equivalence relation if it isReflexiveSymmetricTransitiveWhat does that actually means???Slide14

Equivalence relations

Best to explain by exampleIn the universe of people, xRy if x,y have the

same birthdayRelfexive: xRx Symmetric: xRy implies yRxTransitive: if x,y have the same birthday, and y,z have the same birthday, then so do x,z.14Slide15

Equivalence relations

An equivalence relation partitions the universe to equivalence classesE.g. all people who were born on 1/1/2011 is one equivalence class

Reflexive: a person has the same birthday as himself… dahhh….Symmetric: if x,y have the same birthday then so do y,xTransitive: if x,y have the same birthday, and y,z have the same birthday, then so do x,z15Slide16

Equivalence relations

As a graph16Slide17

Equivalence relations

As a graph17

Equivalence classesSlide18

Equivalence relations

Which of the following is an equivalence relation in the universe of integer numbersx divides yx*y>0x+y>0

x+y is evenNone/more than one/other18Slide19

Equivalence relations

Which of the following is an equivalence relation in the universe of graphsx,y have the same number of verticesx,y have the same edges

x,y are both Eulerianx,y are the same up to re-labeling the vertices (isomorhpic)None/more than one/other19Slide20

Equivalence relations as functions

We can see an equivalence relation as a function Universe

Property E.g. People birthday Integers sign Graphs #verticesAn equivalence class is the set of elements mapped to the same value20