2. Chapter 2
Sets
Homework 3
Given U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } as a set universe and the sets :
A = { 1, 2, 3, 4, 5 },
B = { 4, 5, 6, 7 },
C = { 5, 6, 7, 8, 9 },
D = { 1, 3, 5, 7, 9 },
E = { 2, 4, 6, 8 },
F = { 1, 5, 9 }.
Determine:
A  C
A  B
A  F
(C  D)  E
(F – C) – A

3. Chapter 2
Sets
Solution to Homework 3
Given U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } as a set universe and the sets :
A = { 1, 2, 3, 4, 5 },
B = { 4, 5, 6, 7 },
C = { 5, 6, 7, 8, 9 },
D = { 1, 3, 5, 7, 9 },
E = { 2, 4, 6, 8 },
F = { 1, 5, 9 }.
Determine:
A  C = { 1,2,3,4,5,6,7,8,9 } = U
A  B = { 4,5 }
A  F = { 6,7,8,9 }  { 1,5,9 } = { 9 }
(C  D)  E = { 1,3,6,8 }  { 2,4,6,8 } = { 1,2,3,4 }
(F – C) – A = { 1 } – { 1,2,3,4,5 } = 

4. Cardinality of the Set Union
Chapter 2
Sets
Cardinality of the Set Union
For any sets A and B, the following set equation apply:
A  B = A + B – A  B
A  B = A + B – 2A  B
A  B
A  B

5. Example 1: Cardinality of the Set Union
Chapter 2
Sets
Example 1: Cardinality of the Set Union
Example: In a poll among 40 students, it comes out that 32 of them prefer Internet Explorer, 18 students like Mozilla Firefox more, and 2 students do not like either of the browsers. Determine:
The number of students who like Internet Explorer or Mozilla Firefox.
The number of students who like Internet Explorer or Mozilla Firefox, but not both of them.
Solution:
Define U = { The number of students participating in the poll}
A = { The number of students who like Internet Explorer }
B = { The number of students who like Mozilla Firefox }
Then
U= 40, A= 32, B= 18, A  B= 2

6. Example 1: Cardinality of the Set Union
Chapter 2
Sets
Example 1: Cardinality of the Set Union
The number of students who like Internet Explorer or Mozilla Firefox.
A  B = U – A  B = 40 – 2 = 38
The number of students who like Internet Explorer or Mozilla Firefox, but not both of them.
A  B = A + B – A  B = – 38 = 12
A  B = A + B – 2A  B= – 212 = 26
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7. Cardinality of the Set Union
Chapter 2
Sets
Cardinality of the Set Union
For any sets A, B, and C the following set equation apply:
A  B  C = A + B + C – A  B – A  C –
B  C + A  B  C

8. Example 2: Cardinality of the Set Union
Chapter 2
Sets
Example 2: Cardinality of the Set Union
Example: Among integers between (and including) 101 and 600, how many numbers are not divisible by 4 and not also by 5 or divisible by 4 and 5?
Solution: Define U = { The number of integers between (and including) 101 and 600 } A = { Members of U, divisible by 4 } B = { Members of U, divisible by 5 }
Then
U= 500
A= 500/4 = 125
B= 500/5 = 100
A  B = 500/20 = 25

9. Example 2: Cardinality of the Set Union
Chapter 2
Sets
Example 2: Cardinality of the Set Union
Then
U= 500
A= 500/4 = 125
B= 500/5 = 100
A  B = 500/20 = 25
A  B = A + B – 2A  B
= – 225
= 175
A  B = U – A  B
= 500 – 175
= 325 |

10. Chapter 3
Relation and Function
Relations
A relation R from set A to set B is a subset of the Cartesian product A  B.
Notation: R  (A  B)
a R b is the notation for (a,b)  R, with the meaning “relation R relates a with b.”
a R b is the notation for (a,b)  R, with the meaning “relation R does not relate a with b.”
Set A is denoted as the domain of R. Set B is denoted as the range of R.

11. Chapter 3
Relation and Function
Relations
Example: Given A = { Amir, Budi, Cora } B = { Discrete Mathematics (DM), Data Structure and Algorithm (DSA), State Philosophy (SP), English III (E3) }
A  B = { (Amir,DM), (Amir, DSA), (Amir,SP), (Amir,E3), (Budi,DM), (Budi, DSA), (Budi,SP), (Budi,E3), (Cora,DM), (Cora, DSA), (Cora,SP), (Cora,E3) }
Suppose R is a relation that describes the subjects taken by a student in a certain semester, where: R = { (Amir,DM), (Amir, SP), (Budi,DM), (Budi,E3), (Cora,SP) }
It can be seen that:
R  (A  B)
A is the domain of R, B is the range of R
(Amir,DM)  R or Amir R DM
(Amir,DSA)  R or Amir R DSA

12. Relations Example: Take P = { 2,3,4 } Q = { 2,4,8,9,15 }
Chapter 3
Relation and Function
Relations
Example: Take P = { 2,3,4 } Q = { 2,4,8,9,15 }
If the relation R from P to Q is defined as:
(p,q)  R if p is the factor of q,
then the followings can be obtained:
R = { (2,2),(2,4),(2,8),(3,9),(3,15),(4,4),(4,8) }.

13. Representation of Relations
Chapter 3
Relation and Function
Representation of Relations
Arrow Diagrams

14. Representation of Relations
Chapter 3
Relation and Function
Representation of Relations
Tables

15. Representation of Relations
Chapter 3
Relation and Function
Representation of Relations
Matrices
Suppose R is a relation between A = { a1,a2, …,am } and B = { b1,b2, …,bn }, then the relation R can be presented by the matrix M = [mij].
where:

16. Representation of Relations
Chapter 3
Relation and Function
Representation of Relations
Matrices
a1 = Amir, a2 = Budi, a3 = Cora, and b1 = DM, b2 = DSA, b3 = SP, b4 = E3
p1 = 2, p2 = 3, p3 = 4, and q1 = 2, q2 = 4, q3 = 8, q4 = 9, q5 = 15
a1 = 2, a2 = 3, a3 = 4, a4 = 8, a5 = 9

17. Representation of Relations
Chapter 3
Relation and Function
Representation of Relations
Directed Graph (Digraph)
Relation on one single set can be represented graphically by using a directed graph or digraph.
Digraphs are not defined to represent a relation from one set to another set.
Each member of the set is marked as a vertex (node), and each relation is denoted as an arc (bow).
If (a,b)  R, then an arc should be drawn from vertex a to vertex b. Vertex a is called initial vertex while vertex b terminal vertex.
The pair of relation (a,a) is denoted with an arch from vertex a to vertex a itself. This kind of arc is called a loop.

18. Representation of Relations
Chapter 3
Relation and Function
Representation of Relations
Directed Graph (Digraph)
Example: Suppose R = { (a,a),(a,b),(b,a),(b,c),(b,d),(c,a),(c,d),(d,b) } is a relation on a set { a,b,c,d },
then R can be represented by the following digraph:

19. Chapter 3
Relation and Function
Binary Relations
The relations on one set (from one member to another member in the same set) is also called binary relation.
A binary relation may have one or more of the following properties:
Reflexive and Irreflexive
Transitive
Symmetric
Anti-symmetric

20. Binary Relations Reflexive and Irreflexive
Chapter 3
Relation and Function
Binary Relations
Reflexive and Irreflexive
Relation R on set A is reflexive if (a,a)  R for each a  A.
Relation R on set A is not reflexive if there exists a  A such that (a,a)  R.
Relation R on set A is irreflexive if (a,a)  R for each a  A.
Relation R on set A is not irreflexive if there exists a  A such that (a,a)  R.
Example: Suppose set A = { 1,2,3,4 }, and a relation R is defined on A, then:
R = { (1,1),(1,3),(2,1),(2,2),(3,3),(4,2),(4,3),(4,4) } is reflexive because there exist members of the relation with the form (a,a) for each possible a, namely (1,1), (2,2), (3,3), and (4,4).
R = { (1,1),(2,2),(2,3),(4,2),(4,3),(4,4) } is not reflexive since (3,3)  R, and also not irreflexive because (1,1), (2,2), and (3,3)  R.

21. Chapter 3
Relation and Function
Binary Relations
If a relation is reflexive, then the digraph is characterized by the loop on each vertex.

22. Chapter 3
Relation and Function
Binary Relations
Example: Given a relation “divide without remainder” for a set of positive integers, is the relation reflexive or not?
Each positive integer can divide itself without remainder  (a,a)  R for each a  A  the relation is reflexive
Example: Given two relations on a set of positive integers N:
S : x + y = 4, T : 3x + y = 10 Are S and T reflexive or not? Irreflexive or not?
S is not reflexive, because although (2,2) is a member of S, there exist (a,a)  S for a  N, such as (1,1), (3,3), ....
S is not irreflexive because (2,2) is a member of S.
T is not reflexive because there is even no single pair (a,a)  T that can fulfill the relation.
T is thus irreflexive.

23. Binary Relations Transitive
Chapter 3
Relation and Function
Binary Relations
Transitive
Relation R on set A is transitive if (a,b)  R and (b,c)  R, then (a,c)  R for all a, b, c  A.

24. Chapter 3
Relation and Function
Binary Relations
Example: Suppose A = { 1, 2, 3, 4 }, and a relation R is defined on set A, then:
R = { (2,1),(3,1),(3,2),(4,1),(4,2),(4,3) } is transitive.
R = { (1,1),(2,3),(2,4),(4,2) } is not transitive because (2,4) and (4,2)  R, but (2,2)  R, also (4,2) and (2,3)  R, but (4,3)  R.
R = { (1,2), (3,4) } is transitive because there is no violation against the rule { (a,b)  R and (b,c)  R }  (a,c)  R.
Table for part a.

25. Chapter 3
Relation and Function
Binary Relations
Example: Is the relation “divide without remainder” on a set of positive integers transitive or not?
It is transitive. Suppose that a divides b without remainder and b divides c without remainder, then certainly a divides c without remainder.
{ a R b  b R c }  a R c
Example: Given two relations on a set of positive integers N:
S : x + y = 4, T : 3x + y = 10
Are S and T transitive or not?
S is not transitive, because i.e., (3,1) and (1,3) are members of S, but (3,3) and (1,1) are not members of S.
T = { (1,7),(2,4),(3,1) }  not transitive because (3,7)  R.

26. Binary Relations Symmetric and Anti-Symmetric
Chapter 3
Relation and Function
Binary Relations
Symmetric and Anti-Symmetric
Relation R on set A is symmetric if (a,b)  R, then (b,a)  R for all a,b  A.
Relation R on set A is not symmetric if there exists (a,b)  R such that (b,a)  R.
Relation R on set A such that if (a,b)  R and (b,a)  R then a = b for (a,b)  A, is called anti-symmetric.
Relation R on set A is not anti-symmetric if there exist different a and b such that (a,b)  R and (b,a)  R.
Symmetric
Anti-Symmetric

27. Chapter 3
Relation and Function
Binary Relations
Example: Suppose A = { 1, 2, 3, 4 }, and a relation R is defined on set A, then:
R = { (1,1),(1,2),(2,1),(2,2),(2,4),(4,2),(4,4) }
is symmetric, because if (a,b)  R then (b,a)  R also. Here, (1,2) and (2,1)  R, as well as (2,4) and (4,2)  R.
is not anti-symmetric, because i.e., (1,2)  R and (2,1)  R while 1  2.
R = { (1,1),(2,3),(2,4),(4,2) }
is not symmetric, because (2,3)  R, but (3,2)  R.
is not anti-symmetric, because there exists (2,4)  R and (4,2)  R while 2  4.

28. Chapter 3
Relation and Function
Binary Relations
Example: Suppose A = { 1, 2, 3, 4 }, and a relation R is defined on set A, then:
R = { (1,1),(2,2),(3,3) }
is symmetric and anti-symmetric, because (1,1)  R and 1 = 1, (2,2)  R and 2 = 2, and (3,3)  R and 3 = 3.
R = { (1,1),(1,2),(2,2),(2,3) }
is not symmetric, because (2,3)  R, but (3,2)  R.
is anti-symmetric, because (1,1)  R and 1 = 1 and, (2,2)  R and 2 = 2.

29. Chapter 3
Relation and Function
Binary Relations
Example: Suppose A = { 1, 2, 3, 4 }, and a relation R is defined on set A, then:
R = { (1,1),(2,4),(3,3),(4,2) }
is symmetric.
is not anti-symmetric, because there exist (2,4) and (4,2) as member of R while 2  4.
R = { (1,2),(2,3),(1,3) }
is not symmetric.
is anti-symmetric, because there is no different a and b such that (a,b)  R and (b,a)  R (which will violate the anti-symmetric rule).

30. Chapter 3
Relation and Function
Binary Relations
Example: Show that the relation R = { (1,1),(2,2),(2,3),(3,2),(4,2),(4,4)} is not symmetric and not anti-symmetric.
R is not symmetric, because (4,2)  R but (2,4)  R.
R is not anti-symmetric, because (2,3)  R and (3,2)  R but 2  3.

31. Chapter 3
Relation and Function
Binary Relations
Example: Is the relation “divide without remainder” on a set of positive integers symmetric? Is it anti-symmetric?
It is not symmetric, because if a divides b without remainder, then b cannot divide a without remainder, unless if a = b. For example, 2 divides 4 without remainder, but 4 cannot divide 2 without remainder. Therefore, (2,4)  R but (4,2)  R.
It is anti-symmetric, because if a divides b without remainder, and b divides a without remainder, then the case is only true for a = b. For example, 3 divides 3 without remainder, then (3,3)  R and 3 = 3.

32. Chapter 3
Relation and Function
Binary Relations
Example: Given two relations on a set of positive integers N:
S : x + y = 4, T : 3x + y = 10
Are S and T symmetric? Are they anti-symmetric?
S is symmetric, because take (3,1) and (1,3) are members of S.
S is not anti-symmetric, because although there exists (2,2)  R, but there exist also { (3,1),(1,3) }  R while 3  1.
T = { (1,7),(2,4),(3,1) }  not symmetric. T = { (1,7),(2,4),(3,1) }  anti-symmetric.

33. Chapter 3
Relation and Function
Homework 4
For each of the following relations on set A = { 1,2,3,4 }, check each of them whether they are reflexive, irreflexive, transitive, symmetric, and/or anti-symmetric:
R = { (2,2),(2,3),(2,4),(3,2),(3,3),(3,4) }
S = { (1,1),(1,2),(2,1),(2,2),(3,3),(4,4) }
T = { (1,2),(2,3),(3,4) }
Represent the relation R, S, and T using matrices and digraphs.

34. Chapter 3
Relation and Function
Homework 4A
For each of the following relations on set A = { 1,2,3,4 }, check each of them whether they are reflexive, irreflexive, transitive, symmetric, and/or anti-symmetric:
R = { (1,3),(1,4),(2,2),(4,1),(4,3) }
S = { (1,1),(1,2),(1,3),(1,4),(2,1),(2,2) }
T = { (2,1),(2,2),(3,4),(4,1),(4,4) }
For the set B = { h,i,j,k,l }, with at least 4 members of relation, create:
A relation V which is reflexive and transitive.
A relation W which is irreflexive and symmetric.