CHAPTER (6) RELATIONS. RELATIONS : - Let A and B be sets. A binary relation from A to B is subset of A* B …

CHAPTER (6)

RELATIONS

RELATIONS:

-Let A and B be sets . A binary relation from A to B is subset of

A* B…

EXAMPLE:

Let A be the set of student in your school and let B be the set of

courses , let R be the relation that consists of those pairs (a,b)

A) Fine the pair from this relation B) The pair (arabic ,haya) relation

from(a,b)?! Solution:

(1 )(MONA,ENGLISH.. )(2 )(ARABIC ,HAYA ) is not in R..

INVERSE Relation: Let R be any relation from a set A to set B

,R=(a,b) , R=(b,a) The inverse of R denoted by R

-1-1

EXAMPLE: Let A = [1,2,3] and B=[x,y,z]

Fine the inverse relation… Solution:

[ R=[ (1,y),(1,z),(3,y)

R=[ (y,1),(z,1),(y,3) ]

Directed graphs of relation on sets

-1

EXAMPLE: Draw the directed graph of relation on the

set: A=[1,2,3] (Fine 7 pairs)

Solution: R=[(1,2),(2,2),(2,4),(3,2),(3,4),(4,1),(4,3)]

1 2

3 4

PICTURES OF RELATIONS SETS’’:

Suppose A and B are set , there are two ways of picturing

A relation R from A to B

1 (Matrix of the relation2 (Arrow diagram

EXAMPLE:

Let A=[1,2,3] AND B=[x,y,z]Fine A R B and draw pictures of it (fine 3

pairs) Solution:

R=[(1,y),(1,z),(3,y)]Picture of R

Z Y X A1 1 0 10 0 0 20 1 0 3

MATRIX OF RX

Y

Z

1

2

3ARROW OF DIAGRAM

TYPE OF RELATIONS:

1 -REFLIXUE : X R X 2 -SYMMETRIC : a R b b R a

3 -anti-symmetric : arb ^ brc a=b 4 -transitive : a R b ^ b R a a R b

EXAMPLE:

Given X breather of Y , C breather of Y ,

-WHAT IS THE TYPE OF THIS RELATION?

Solution : X breather Y Y breather C

Then C breather X this transitive relation

EXAMPLE:

Given A=[1,2,3] AND B =[1,2] , R=[(2,1),(3,1),(3,2)]

FINE MATRIX FOR R..

Solution:

MR =

0 0

1 01

1

EXAMPLE:

Let A=[a1,a2,a3] and B=[b1,b2,b3,bn,b5]Which order pairs are in the relation R

represent By matrix

Mr=

0 1 0 0 0

1 0 1 1 0

1 0 1 0 1

Solution:

R=[(A1,B2),(A2,B1),(A2,B3),(A2,B4),(A3,B1),(A3,B3),(A3,B5)]

THANKS FOR ALL