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
-1
-1
The inverse of R denoted by R , R=(a,b) , R=(b,a)
EXAMPLE :
Let A = [1,2,3] and B=[x,y,z]
Fine the inverse relation …
Solution :
R=[ (1,y),(1,z),(3,y) ]
-1
R=[ (y,1),(z,1),(y,3) ]
Directed graphs of relation on sets
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 relation
2) 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
A
1
2
3
X
0
0
0
Y
1
0
1
Z
1
0
0
MATRIX OF R
X
1
2
Y
3
Z
ARROW OF DIAGRAM
TYPE OF RELATIONS :
1- REFLIXUE : X R X
2- SYMMETRIC : a R b
bRa
3- anti-symmetric : arb ^ brc
a=b
4- transitive : a R b ^ b R a
aRb
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
0
1
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