Lecture 4.1: Relations Basics

CS 250, Discrete Structures, Fall 2011

Nitesh Saxena

*Adopted from previous lectures by Cinda Heeren

10/24/2011 Lecture 4.1: Relations Basics

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your disposal for the rest of the semester 10/24/2011 Lecture 4.1: Relations Basics

10/24/2011 Lecture 4.1: Relations Basics

Outline

Relation Definition and Examples Types of Relations Operations on Relations

Lecture 4.1: Relations Basics

Relations

Recall the definition of the Cartesian (Cross) Product:

The Cartesian Product of sets A and B, A x B, is the setA x B = {<x,y> : xA and yB}.

A relation is just any subset of the CP!! R AxB

Ex: A = students; B = courses. R = {(a,b) | student a is enrolled in class b}We often say: aRb if (a,b) belongs to R

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Lecture 4.1: Relations Basics

Relations vs. Functions

Recall the definition of a function:f = {<a,b> : b = f(a) , aA and bB}

Is every function a relation?

Draw venn diagram of cross products, relations, functions

Yes, a function is a special kind

of relation.

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Lecture 4.1: Relations Basics

Properties of Relations

Reflexivity:A relation R on AxA is reflexive if for all aA, (a,a) R.

Symmetry:A relation R on AxA is symmetric if

(a,b) R implies (b,a) R .

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Lecture 4.1: Relations Basics

Properties of Relations

Transitivity:A relation on AxA is transitive if

(a,b) R and (b,c) R imply (a,c) R.

Anti-symmetry:A relation on AxA is anti-symmetric if

(a,b) R implies (b,a) R.

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Lecture 4.1: Relations Basics

Properties of Relations - Techniques

How can we check for the reflexive property?

Draw a picture of the relation (called a “graph”).

Vertex for every element of AEdge for every element of RNow, what’s R?

{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}

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Loops must exist on EVERY

vertex.

10/24/2011

Lecture 4.1: Relations Basics

Properties of Relations - Techniques

How can we check for the symmetric property?

Draw a picture of the relation (called a “graph”).

Vertex for every element of AEdge for every element of R

Now, what’s R?{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}

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EVERY edge must have a return edge.

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Lecture 4.1: Relations Basics

Properties of Relations - Techniques

How can we check for transitivity?

Draw a picture of the relation (called a “graph”).

Vertex for every element of AEdge for every element of RNow, what’s R?

{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}

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34

A “short cut” must be present for EVERY path

of length 2.

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Lecture 4.1: Relations Basics

Properties of Relations - Techniques

How can we check for the anti-symmetric property?

Draw a picture of the relation (called a “graph”).Vertex for every element of AEdge for every element of R

Now, what’s R?{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}

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34

No edge can have a

return edge.

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Lecture 4.1: Relations Basics

Properties of Relations - Examples

Let R be a relation on People,R={(x,y): x and y have lived in the same

country}

NoIs R transitive?

Is it reflexive? Yes

Is it symmetric? Yes

Is it anti-symmetric?

No

1 2

3?

1

?

1 2

?

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Lecture 4.1: Relations Basics

Properties of Relations - Examples

Let R be a relation on positive integers,R={(x,y): 3|(x-y)}

YesIs R transitive?

Suppose (x,y) and (y,z) are in R.

Then we can write 3j = (x-y) and 3k = (y-z) Definition of “divides”

Can we say 3m = (x-z)?

Is (x,z) in R?

Add prev eqn to get: 3j + 3k = (x-y) + (y-z)

3(j + k) = (x-z)

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Lecture 4.1: Relations Basics

Properties of Relations - Techniques

Let R be a relation on positive integers,R={(x,y): 3|(x-y)}

YesIs R transitive?

Is it reflexive? Yes

Is (x,x) in R, for all x?

Does 3k = (x-x) for some k?Definition of

“divides”

Yes, for k=0.

10/24/2011

Lecture 4.1: Relations Basics

Properties of Relations - Techniques

Let R be a relation on positive integers,R={(x,y): 3|(x-y)}

YesIs R transitive?

Is it reflexive? Yes

Is it symmetric? Yes

Suppose (x,y) is in R.

Then 3j = (x-y) for some j.Definition of

“divides”

Yes, for k=-j.

Does 3k = (y-x) for some k?

10/24/2011

Lecture 4.1: Relations Basics

Properties of Relations - Techniques

Let R be a relation on positive integers,R={(x,y): 3|(x-y)}

YesIs R transitive?

Is it reflexive? Yes

Is it symmetric? Yes

Is it anti-symmetric?

No

Suppose (x,y) is in R.

Then 3j = (x-y) for some j.Definition of

“divides”

Yes, for k=-j.

Does 3k = (y-x) for some k?

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Lecture 4.1: Relations Basics

More than one relation

Suppose we have 2 relations, R1 and R2, and recall that relations are just sets! So we can take unions, intersections, complements, symmetric differences, etc.

There are other things we can do as well…

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Lecture 4.1: Relations Basics

More than one relationLet R be a relation from A to B (R AxB), and let S be a relation

from B to C (S BxC). The composition of R and S is the relation from A to C (SR AxC):

SR = {(a,c): bB, (a,b) R, (b,c) S}

SR = {(1,u),(1,v),(2,t),(3,t),(4,u)}

A B C

1

2

3

4

x

y

z

s

t

u

v

R S

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Lecture 4.1: Relations Basics

More than one relationLet R be a relation on A. Inductively define

R1 = RRn+1 = Rn R

R2 = R1R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1), (4,2)}

A A A

1

2

3

4

1

2

3

4

R R1

1

2

3

4

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Lecture 4.1: Relations Basics

More than one relationLet R be a relation on A. Inductively define

R1 = RRn+1 = Rn R

R3 = R2R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1),(4,2),(4,3)}

A A A

1

2

3

4

1

2

3

4

R R2

1

2

3

4 … = R4

= R5

= R6… 10/24/2011

10/24/2011 Lecture 4.1: Relations Basics

Today’s Reading Rosen 9.1