1. Agenda Lecture Content: Relations (Relasi)
Matrices of Relations (Matriks Relasi)
Review Quiz
Exercise

2. Relations

3. Relations Relations generalize the notion of function.
 Function is a special type of Relation
Relationships between elements of sets occur in many contexts.
To express a relationship between elements of two sets: ordered pairs made up of two related elements  binary relations.
(Bogor, Jawa Barat), (Surabaya, Jawa Timur)

4. Relation: Definition
Let A and B be sets. A binary relation from A to B is a subset of A  B.
Cartesian Product
IF R : binary relation THEN
a R b: (a, b)  R
a R b: (a, b)  R
Notation:
domain: {a  A | (a,b)  R for some b  B}
range: {b  B | (a,b)  R for some a  A}

5. Example
A = {0, 1, 2}
B = {a, b}
Relation from a to b: {(0, a), (0, b), (1, a), (2, b)}
0 R a ?
0 R 2 ?
2 R a ?

6. Note: Function is a special type of Relation
Properties:
The domain of the function f is equal to A
For each a  A, there is exactly one b  B such that (a,b)  f.

7. How to represent relation
R = {(0, a), (0, b), (1, a), (2, b)}
Table
A B
0 a
0 b
1 a
2 b

8. How to represent relation
Digraphs = Directed Graphs (chapter 8)
A = {a, b, c, d}, R = {(a, b), (a, d), (b, b), (b, d), (c, a), (c, b), (d, b)}. a b d c

9. How to represent relation1 A 2 3 4 B C D R A B C D 1 2 3 4 1 1 1 1 1 1

10. Relations on a Set
A Relation on the set A is a relation from A to A
 a subset of A  A
A = {1, 2, 3, 4}
Which pairs are belong to R = {(a, b) | a ≤ b} ?
{(1, 1), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4), (2,2), (3,3),(4,4)}

11. Example Ordered pairs: (1, 1), (1, 2), (2, 1), (1, -1) and (2, 2)
R1 = {(a, b) | a ≤ b}
R2 = {(a, b) | a > b}
R3 = {(a, b) | a = b}
R4 = {(a, b) | a = b or a = -b}
R5 = {(a, b) | a = b + 1}
R6 = {(a, b) | a + b ≤ 3}

12. Properties of Relations
Reflexive
Symmetric
Antisymmetric
Transitive
Composite

13. Reflexive
A relation R on a set A is called reflexive if (a, a)  R for every element a  A.
R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
R2 = {(1, 1), (1, 2), (2, 1)}
R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}
R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
R6 = {(3, 4)}

14. Which one is reflexive?
R1 = {(a, b) | a ≤ b}
R2 = {(a, b) | a > b}
R3 = {(a, b) | a = b}
R4 = {(a, b) | a = b or a = -b}
R5 = {(a, b) | a = b + 1}
R6 = {(a, b) | a + b ≤ 3}

15. Symmetric and Antisymmetric
A relation R on a set A is called symmetric: if (a, b)  R then (b, a)  R.
A relation R on a set A is called antisymmetric: if for all a, b  R, IF (a, b)  R AND a ≠ b THEN (b, a)  R .
(IF (a, b)  R AND (b, a)  R THEN a = b )
R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
R2 = {(1, 1), (1, 2), (2, 1)}
R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}
R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
R6 = {(3, 4)}

16. Which one is symmetric, antisymmetric?
R1 = {(a, b) | a ≤ b}
R2 = {(a, b) | a > b}
R3 = {(a, b) | a = b}
R4 = {(a, b) | a = b or a = -b}
R5 = {(a, b) | a = b + 1}
R6 = {(a, b) | a + b ≤ 3}

17. Transitive
A relation R on a set A is called transitive if whenever (a, b)  R and (b, c)  R then (a, c)  R, for all a, b, c  A .
R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
R2 = {(1, 1), (1, 2), (2, 1)}
R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}
R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
R6 = {(3, 4)}

18. Which one is transitive?
R1 = {(a, b) | a ≤ b}
R2 = {(a, b) | a > b}
R3 = {(a, b) | a = b}
R4 = {(a, b) | a = b or a = -b}
R5 = {(a, b) | a = b + 1}
R6 = {(a, b) | a + b ≤ 3}

19. S  R = {(a,c) | (a,b)  R and (b,c)  S for some b  B}
Composition
Let R be a relation from a set A to a set B and S a relation from B to a set C. We denote the composition of R and S by S  R.
S  R = {(a,c) | (a,b)  R and (b,c)  S for some b  B}
R is the relation from {1, 2, 3} to {1, 2, 3, 4}
R = {(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)}
S is the relation from {1, 2, 3, 4} to {0, 1, 2}
S = {(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)}
S  R ?
S  R = {(1, 0), (1, 1), (2, 1), (2, 2), (3, 0), (3, 1)}

20. Digraph and Properties of Relations
Reflexive: there is a loop at every vertex.
Symmetric: whenever there is a directed edge from x to y, there is also a directed edge from y to x.
Antisymmetric: whenever there is a directed edge from x to y, and x ≠ y, then there is no directed edge from y to x.
Transitive: whenever there are directed edges from x to y and from y to z, then there is also a directed graph from x to z.

21. Partial Orders Partial Orders is a relation R that is
reflexive, antisymmetric and transitive.
Ex.
R1 = {(a, b) | a ≤ b}

22. Equivalence Relations
A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive
Ex.
R = {(1,1), (1,3), (1,5), (2,2),(2,4),(3,1),(3,3), (3,5),(4,2),(4,4),(5,1),(5,3),(5,5)}
R is an equivalence relation on {1,2,3,4,5}

23. Matrices of Relations

24. Matrices of Relations
Alternative methods for representing relations:
Zero – One matrices

25. Zero-One Matrices
A = {a1, a2, …, an}
B = {b1, b2, …, bn}
R is a relation from A to B.
R can be represented by the matrix MR = [mij] where
mij = 1, if (ai,bj)R, and
mij = 0, if (ai,bj)R.

26. Example
A = {1, 2, 3}
B = {1, 2}
R = {(a, b) | a > b}
R = {(2, 1), (3, 1), (3, 2)}

27. Example
A = {a1, a2, a3 , a4}
B = {b1, b2, b3 , b4}
R ?

28. Zero-One Matrices Find Zero-One Matrices for: Reflexive Relation ?
Symmetric Relation ?
Antisymmetric Relation ?

29. Antisymmetric Relation ?
Zero-One Matrices
Reflexive Relation ?
Symmetric Relation ?
Antisymmetric Relation ?

30. Combining Relations A = {1, 2, 3} B = {1, 2, 3, 4}
R1  R2 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (3, 3)}
R1  R2 = {(1, 1)}

31. Combining Relations

32. Combining Relations

33. Review Quiz Exercise