1. CHAPTER (6) RELATIONS

2. RELATIONS : - Let A and B be sets. A binary relation from A to B is subset of A* B …

3. 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..

4. 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

5. 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

6. 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)] 12 34

7. 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

8. 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

9. ZYXA 1101 0002 0103 MATRIX OF R XYZXYZ 123123 ARROW OF DIAGRAM

10. 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

11. 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

12. EXAMPLE : Given A=[1,2,3] AND B =[1,2], R=[(2,1),(3,1),(3,2)] FINE MATRIX FOR R.. Solution : MR = 0 1 0 1

13. 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

14. Solution : R=[(A1,B2),(A2,B1),(A2,B3),(A2,B4),(A3,B1),(A3,B3),(A3, B5)]

15. THANKS FOR ALL