RelationsChapter 3

Relations

DefinitionsRelations

Binary relationsDomainRangeVertexDirect edgeLoopsReflective RelationsSympatric RelationsAnti Symmetric Relations

Metrics of RelationsReflective RelationsSymmetric RelationsMatrix multiplications

RelationsA set of ordered pairs.The existence or the presence of the order

pair (a, b) indicates that there is a relationship from a to b.The relational Dbase among records in a data

base is based on the Relations concept.

A relation from one set to another can be thought of as which element of the first set is related to which element in the second set

Student Course

Bill Computer Science

Mary Math

Bill Art

Beth Beth

HistoryComputer Science

Dave Math

Student col is the Domain, Course Col is the Range

Binary Relation R A binary relation R from a set X to a set Y is subset of the Cartesian product X*

Y If (x, y) ∈ R, we write x R y and we say x related to y. If x = y we say R is Binary relation on X. A function is a special type of Relation, a function f from X to Y is a Relation

from X to Y with the properties: The Domain f equals X For each x ∈ X, there is exactly y ∈ Y such that (x,y) ∈ f

If X = { Bill, Mary, Beth, Dave} and Y = {Computer Science, Math, Art, History}Then Our Relation from the previous table R = { (Bill, Computer Science), (Mary, Math),(Bill, Art) (Beth, History,) , (Beth, Computer Science ),(Dave. Math)}.

Since the order pair (Beth, History) ∈ R, we can write Beth R History

This shows that a Relation can b given by specifying which ordered pair belongs to the Relation

Defining a Relation by giving a Rule for membership in the Relation

X = {2, 3, 4} an Y = {3, 4, 5, 6, 7}If we define a relation R from X to Y by (x, y) ∈ R if x

divides by y.We obtain

R ={(2,4), (2,6), (3,3), (3,6), (4,4)And if we Write R as a table we obtain

X Y

2 4

2 6

3 3

3 6

4 4

The Domain of R is the set {2,3,4}

The Range of the set R is{3, 4, 6}

GraphsLet R be the Relation on X = {1,2,3,4}

defined by

the order pair (x, y) ∈ R if x ≤ y and x, y ∈ XThe Relation R can be written as R = {(1,1),(1,2,),(1,3), (1,4), (2,2),(2,3),(2,4), (3,3),(3,4),(4,4)}

The Relation R can be drawn as a graph, we call it a digraph (direct graph and will be covered in Chapter 8 1

3 4

2 1, 2, 3, 4 are called verticesThe Arrows are called edgesIf there is a relation between vertices we draw an arrow to represent the relation

Element(vertices) with the form off (x, x) are called lops

Digraph of RelationThe Relation R on X = {a, b, c, d} given by

the DigraphR = {(a,b),(a,d),(d,c), (c,d), (b,b)}

a b

cd

Reflective relation RA Relation R on the set X is called reflective if

(x, x) ∈ R for every x ∈ XThe relation R on the set X ={1,2,3,4}is reflective for each x ∈ X, (x, x) ∈ RSpecifically: {(1,1,), (2,2), (3,3),(4,4)} are each in R

The Digraph of reflective Relation has a loop on every vertex

THE GRAPH ON SLIDE 8 IS REFLEXSIVE

Not a reflective The Relation R R = {(a,b),(a,d),(d,c), (c,d),

)} on X = {a, b, c, d} given by the Digraph

a b

cd

Is not a reflective relation because none of the Vertices have a loop, there is no (b,b), (c,c).. (b,b) and (c,c) ∈ R

Not a reflective The Relation R on X = {a, b, c, d} given by

the DigraphR = {(a,a),(b,c),(c,b), (d,d), )} Is not a reflective relation is b ∈ X, c ∈ X

But there is no loop on (b,b), (c,c) Vertices have a loop a d

cb

No LoopNo

Loop

symmetricA relation R on set X is called Symmetric if for all x, y ∈ X , if (x, y) ∈ R then (y, x) ∈ R The Relation R = {(a,a), (b,c),(c,b), (d,d) } on X = {a, b, c, d} Is symmetric because for all x, y if (x,y) ∈ R then (y, x) ∈ R

example ((b,c) is in R and (c,b) is in also in R The diagraph of a symmetric relation has the property that

whenever there is a direct edge from v to w there is also a direct edge from w to v

b c

a d

NOT Symmetric exampleThe Relation R on X = {1,2,3,4} defined by

(x, y) ∈ R if x ≤ y , x,y ∈ X R = {(1,1),(1,2,),(1,3), (1,4), (2,2),(2,3),(2,4), (3,3),(3,4),(4,4)}

is Not Symmetric.(2,3) ∈ R but (3,2) ∈ R.The digraph of this relation has a direct edge from 2 to 3 but no direct edge between 3 and 21

3

2

4

AntiSymmetric

The relation R on X is antisymmetric if for all x,y ∈ X, if (x,y) ∈ R and (y, x) ∈ R then x =y

Example:The relation R on X = {1,2,3,4} defined by (x ,y)

∈ R if x ≤ y, x,y ∈ X is antisymmetric because for all X,Y , IF (x,y) ∈ R and (y, x) ∈ R

x = yWe also can say if (x, y) ∈ R and (y, x) ∈ R then x = y.

AntisymmetricIf a relation R has no members of the form (x,

y), x ≠ yThe relation is called Anti Symmetric for all x, y ∈ X , if x ≠ y then (x, y) ∈ R or (y, x) ∈ R

Example R ={(a, a),(b, b),(c, c) on X = {a, b, c} is an antisymmetric

a d

c

Also we can say, the digraph of antisymmetic relation has at most one directed edge between any two distinct vertices

This gra

ph is

also sy

mmetric

and Reflexiv

e

Metrics of RelationsA matrix is a convent way to represent a relation.R from X to Y, such a representation can be used

by computers to a analyze a relation.Row elements of X are labeled in some order and

the column elements of Y are labeled .

Entries in Row X and Column Y are set to 1 if

there is R from X to Y otherwise it is set to 0This type of matrix is called Matrix of relations

from X to Y

Example.

The matrix of the Relation R whenR ={(1, b),(1, d), (2, c), (3, c),(3, b),(4, a)}From X={1, 2,3,4} to Y ={a, b, c, d} IS:

a b c d 1 0 1 0 1 2 0 0 1 0 3 0 1 1 0 4 1 0 0 0The matrix is dependent on the ordering of x, y.How would the Matrix for the Relation R look like if:X= {4, 3,2, 1} and Y = {d, c, b, a} … next slide

R ={(1, b),(1, d), (2, c), (3, c),(3, b),(4, a)}From X={4,3,2,1} to Y ={d, c, b, a} IS:

d c b a 4 0 0 0 1 3 0 1 1 0 2 0 1 0 0 1 1 0 1 0

Example 2

The matrix of the Relation R fromX ={2,3,4} to Y={5,6,7,8}is defined by X R Y if x divides y Note: no remainder

5 6 7 8 2 0 1 0 1 3 0 1 0 0 4 0 0 0 1

Reflective RelationsThe matrix of the RelationR = {(a,a), (b, b), (c, c), (d, d),(b, c), (c, b) } on {a, b, c, d} relative to the ordering {a, b, c, d}

a b c d a 1 0 0 0 b 0 1 1 0 c 0 1 1 0 d 1 0 1 1

Matrix diagonal

• The Relation R is Reflective if and ONLY if A has only ones on the main diagonal .

• The main diagonal is the line from the upper left corner to the lower right corner.

• In other words the Relation R is Reflective if and only if

• (x, x) ∈ R for every x ∈ R.

Symmetric RelationIf the Entry of R,C = the entry of C,R then the

relation is symmetric, otherwise it is anti-symmetric.

We can quickly determine if the relation is symmetric by checking the main diagonalLook at all entries for all R,C or (i, j)Look at all entries for all C,R or (j,i)

If they are all 1’s then it is Symmetric

Based on the above: for the previous example the Relation R is Symmetric

How Matrix Multiplication Relate to the composition of Relations and how can we test for Transitivity by using Matrix Relations?

What is TransitivityTransitivity: Transitive means whenever a

none Zero entry in A ( entry (i, j) ≠ 0) there

is also a none Zero in A2 ( entry (i, j) ≠ 0)

Example Let R be the Relation from X to YX ={1, 2,3} to Y ={a, b} defined by R1 ={(1,a),(2,b), (3,a), (3,b) and R2= the Relation from Y to Z where Z ={x, y, z} And the Relation is defined R2 ={(a,x), (a,y), (b,y), (b,z)}

The matrix for R1 Relative to the ordering of 1,2,3 and a,b

a b 1 1 0 2 0 1 3 1 1

x y z a 1 1 0 b1 1 1

The Matrix of R2 relative to the ordering of (a,b) on (x, y, z)

A1 =

A2 =

1 1 0 A1.A2 = 0 1 1

1 2 1

TransitivityIF A is the matrix of R (relative to some

ordering)

We compute A2

Then we compare A and A2

The relation R is Transitive if and only if whenever an entry i,j in A is a

non zero, an entry i,j in A^2 is also a none zero.

Determine if the Relation R is transitiveThe matrix of Relation: R = {(a,a),(b,b),(c,c),(d,d), (b,c),(c.b)}On. (a,b,c,d) relative to the order a, b, c, d a b c d a b c d 0 0 0 1 1 0 0 0 0 1 1 0 0 2 2 0 0 1 1 0 0 2 2 0 0 0 0 1 0 0 0 1

abcd

abcd

A= A^2 =

Every none Zero i,j in A is matched with a none zero entry in i,j in A^2 Transitive Relation R

R2 O R1To obtain the Matrix of the Relation R2 O R1 Compute the Matrix product A1*A2 Change all none zero entries to 1

Adjacency MatrixWill be covered in chapter 11