Agenda
• Deskripsi perkuliahan Matematika Diskrit
• Topik Minggu 1:
– Himpunan
– Operasi-operasi pada Himpunan
• Pembagian Kelompok dan Latihan Soal
Deskripsi Perkuliahan
• Referensi materi:
– GBPP (Garis-Garis Besar Program Pembelajaran) TI 101
Matematika Diskrit
• Tujuan instruksional:
– Mahasiswa memiliki pengetahuan, pemahaman, dan
kemampuan untuk menerapkan dasar-dasar ilmu Matematika
Diskrit dalam berbagai bidang kehidupan, khususnya di
bidang ICT (Teknik Informatika, Sistem Komputer, dan
Sistem Informasi)
• Mekanisme perkuliahan:
– Paparan teori: 2x50 menit
– Latihan soal (kelompok/individu): 50 menit
– Paparan dalam bahasa Indonesia, slide presentasi dalam
bahasa Inggris (menyesuaikan dengan textbook)
Textbook
R. Johnsonbaugh, Discrete Mathematics, Pearson International
Edition (7th edition), 2009
K.H. Rosen, Discrete Mathematics and Its Applications,
McGraw-Hill (6th edition),2006
S. Lipschutz, et. al, Schaum’s Outline of Theory and Problems
of Discrete Mathematics, McGraw-Hill (3th edition), 2009
Ketentuan Penilaian
• Bobot masing-masing penilaian:
– Tugas Mandiri
– Ujian Tengah Semester (UTS)
– Ujian Akhir Semester (UAS)
30%
30%
40%
• Tugas Mandiri
– Quiz (minimal 2 kali)
• Diumumkan seminggu sebelumnya
– Penyelesaian soal-soal latihan (kelompok/individu)
Disiplin Perkuliahan
• Peringatan dari manajemen universitas tentang
Kehadiran
– Maksimum absensi: 3x dari 14x pertemuan
• Siswa dapat gagal mengikuti UAS apabila prosentase
kehadiran tidak terpenuhi
– Toleransi keterlambatan 15 menit
• Presensi diambil pada 15 menit awal perkuliahan
• Diperiksa kembali di akhir perkuliahan dengan validasi di
euis.umn.ac.id (bersama dengan ketua/wakil ketua kelas)
• Mahasiswa yang terlambat dan dianggap absen tetap dapat
mengikuti perkuliahan
• Apabila mahasiswa ada keperluan mendadak, hubungi
dosen/BAAK untuk memperoleh dispensasi
Kenapa Mahasiswa ICT perlu belajar
Discrete Mathematics (Matematika Diskret)
• Discrete mathematics is mathematics that deal with
discrete objects  objects which are separated from
each other, e.g. integers, propositions,sets,relations,
functions,graphs, etc
• Foundation for ICT applications, e.g. :
– Analysis of algorithms
– Circuit design
– Unicast & multicast routing
– Computer security
– Database
– Deadlock analysis
…
Pokok Bahasan (1)
Minggu ke
Topik
1
-
Konsep Himpunan
Operasi-operasi pada himpunan
2
-
Proposisi
Operator logika dan tabel kebenaran
Implikasi dan bi-implikasi
Tautologi, kontradiksi, dan kontingensi
Argument dan aturan inferensi
Quantifiers
3
-
Pembuktian langsung dan tidak langsung
Metode pembuktian lainnya
Strategi pembuktian
Induksi Matematika
4
-
Fungsi
Barisan dan strings
Deret jumlahan dan kalian
Pokok Bahasan (2)
Minggu ke
Topik
5
-
Relasi
Matriks relasi
6
-
Sistem bilangan Matematika
Sistem bilangan biner, octal, dan hexadecimal
Divisors dan prime numbers
Algoritma Euclid
7
-
Dasar metode perhitungan
Permutasi dan kombinasi
Ujian Tengah Semester (UTS)
Pokok Bahasan (3)
Minggu ke
Topik
8
-
Peluang diskret
Koefisien binomial
Identitas kombinatorik
9
-
Algoritma rekursif
Relasi rekurensi
Penyelesaian relasi rekurensi
Fungsi pembangkit
10
-
Graf dan terminologi graf
Path dan cycle
Euler cycle dan Hamiltonian cycle
11
-
Representasi graf
Graf isomorfis
Graf planar
Permasalahan lintasan terpendek
Pokok Bahasan (4)
Minggu ke
Topik
12
-
Terminologi dan karakterisasi pohon
Spanning Trees
Binary Trees
13
-
Tree traversals
Decision trees
Pohon isomorfis
Game trees
14
-
Kombinatorial sirkuit
Aljabar Boolean
Ujian Akhir Semester (UAS)
Set (Himpunan)
Set
Definition:
A set is unordered collection of objects.
The objects in a set are called the elements, or
members, of the set. A set is said to contain its
elements.
A = { a, b, c, d }
B = { 1, 2, 3 }
Notation:
a is an element of the set: a  A
f is not an element of the set: f  A
Describing a Set (1)
{…}
V is a set of all vowel in alphabet:
{a, e, i, o, u}
O is a set of odd positive integers less than 10:
{1, 3, 5, 7, 9}
P is a set of positive integer less than 100:
{1, 2, 3, …, 99}
D is a set of personal data:
{Joko, student, UMN}  unrelated elements
Describing a Set (2)
Set Builder
O is a set of odd positive integers less than 10:
O = {x | x is an odd positive integer less than 10}
OR
O = {x  Z+ | x is odd and x < 10}
Z+ is the set of all positive integers
Set Builder: Describe the following sets
{0, 3, 6, 9, 12}
{-3, -2, -1, 0, 1, 2, 3}
{m, n, o, p}
{1, 2, 4, 8, 16}
Describing a Set:
Common Notations
Z = {…, -2, -1, 0, 1, 2, …} , the set of integers.
Z+ = {1, 2, 3, …} , the set of positive integers.
(N = {1, 2, 3, …} , the set of natural numbers)
Znonneg ={0}  Z+ = {0, 1, 2, 3, …} , the set of nonnegative integers.
Q = {p/q | p  Z, q  Z, and q ≠ 0} , the set of rational
numbers}.
Q+ is the set of all positive rational numbers.
Q+ = {x  R | x = p/q, for some positive integers p and
q}.
R = the set of real numbers, consisting of all point in a
straight line.
Describing a Set (3)
Venn Diagram
Rectangle: the universal set U
Circle : sets, e.g. V
U
•a
•o
•e
V
•i
•u
Equality
Definition:
Two sets are equal if and only if they have the same
elements. That is, if A and B are sets then A and B
are equal if and only if
For every x, if x  A, then x  B
For every x, if x  B, then x  A
A=B
Examples:
A = {1, 3, 5}
B = {5, 1, 3}
C = {1, 1, 3, 3, 5}
A = B ?  Yes
B = C ?  Yes
Equality
A = { X | X2 + X – 6 = 0 }
B = { 2, 3 }
A=B?
No
A≠B
A = { X | X2 + 5 X + 6 = 0 }
B = { -2, -3 }
A=B?
Yes
Empty Set & Singleton
Definition:
Empty set or null set is a set that has no elements.
Notation:
, { }
Ex. The set of all positive integers that are greater
than their squares.
Singleton set is a set with one element.
Ex.
the set of positive odd integers less than 3:
{1}
Subset
Definition:
The set A is said to be a subset of B if and only if
every element of A is also an element of B.
Notation:
AB
Means: for every x, if x  A then x  B
U
A
B
Subset: Theorem
(i)   S
(ii) S  S
Proof:
Show that: for every x, if x   then x  S)
Because the empty set contains no elements, it
follows that x   is always false.
It means that x    x  S is always true
(you will learn the truth table for conditional
statements in next week’s lecture)
Proper Subset
Definition:
The set A is a subset of B but A ≠ B
Notation:
AB
Means: A is a subset of B and A does not equal B
Note : proper subset pasti subset tetapi subset
belum tentu proper subset.
Subset: Examples
Give examples of:
A  B and B  C
A  B and B  C
Equality of two sets
Definition:
IF A  B and B  A THEN A = B
Example:
A= { , {a}, {b}, {a, b}}
B = {x | x is a subset of the set {a, b}}
A=B
Finite Set and Cardinality
Definition:
Let S be a set. If there are exactly n distinct elements
in S where n is a nonnegative integer, we say that
S is a finite set and that n is the cardinality of S.
Notation:
|S|
A is a set of odd positive integers less than 10.
|A| = 5
S is a set of letters in alphabet.
|S| = 26
B = {1, 1, 3, 5, 7}
|B|=4
Infinite Set
Definition:
A set S is said to be infinite if it is not finite.
Example:
the set of positive integers.
Power Set
Definition:
Given a set S, the power set of S is the set of all
subsets of the set S.
Notation:
P(S)
What is the power set of the set {0, 1, 2}?
P({0, 1, 2}) = {, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1,
2}}
Cardinality of Power Set
|S| = 3
|P(S)| = 23 = 8
Set Operations
(Operasi-operasi pada Himpunan)
Union
Definition:
Let A and B be sets. The union of the sets A and B,
denoted by A  B, is the set that contains those
elements that are either in A or in B, or in both.
Notation:
A  B = {x | x  A or x  B}
Union: Example
A = {1, 3, 5}
B = {1, 2, 3}
A  B = {1, 2, 3, 5}
Intersection
Definition:
Let A and B be sets. The intersection of the sets A
and B, denoted by A  B, is the set containing
those elements in both A and B
Notation:
A  B = {x | x  A and x  B}
Intersection: Example
A = {1, 3, 5}
B = {1, 2, 3}
A  B = {1, 3}
Difference
Definition:
Let A and B be sets. The difference of A and B,
denoted by A – B, is the set containing those
elements that are in A but not in B. it is also called
the complement of B with respect to A.
Notation:
A – B = {x | x  A and x  B}
Difference: Example
A = {1, 3, 5}
B = {1, 2, 3}
A–B={5}
B–A={2}
In general : A – B ≠ B – A
Disjoint
Definition:
Two sets are called disjoint if their intersection is the
empty set.
Notation:
A B=
A = {1, 3, 5}
B = {2, 4, 6}
AB=
Cardinality of Union
Definition:
The cardinality of a union of two finite sets A and B
is | A  B |
Notation:
|AB|=|A|+|B|-|A B|
Complement
Definition:
Let U be the universal set. The complement of the
set A, denoted by Ā, is the complement of A with
respect to U or U – A.
Notation:
Ā = U – A ={x | x  A}
Example
A = {a, e, i, o, u}
U – A = all other alphabets
Let A be the set of positive integers greater than 10.
U is the set of all positive integers.
U – A = { 1, 2, 3, …, 10}
Example
A group of 165 students.
8 are taking calculus, computer science and
psychology
33 are taking calculus, computer science
20 are taking calculus and psychology
24 are taking computer science and psychology
79 are taking calculus
83 are taking psychology
72 are taking computer science
Example
A group of 165 students.
8 are taking calculus, computer science and
psychology
25(33-8) are only taking calculus, computer science
12(20-8) are only taking calculus and psychology
16(24-8) are only taking computer science and
psychology
34(79-25-12-8) are only taking calculus
47(83-12-16-8) are only taking psychology
23(72-25-16-8) are only taking computer science
Set Identities Table
Ordered n-tuple
Definition:
The ordered n-tuple (a1, a2, …, an) is the ordered
collection that has a1 as its first element, a2 as its
second element, …, and an as its nth element.
Two ordered n-tuples are equal if and only if each
corresponding pair of their elements is equal.
(a1, a2, …, an) = (b1, b2, …, bn) if and only if ai=bi
Cartesian Products
Definition:
Let A and B be sets. The Cartesian product of A and
B, denoted by A  B, is the set of all ordered pairs
(a, b), where a  A and b  B.
Notation:
A  B = {(a,b) | a  A  b  B}
Cartesian Products: Example
A = {0, 1}
B = {1, 2}
C = {0, 1, 2}
ABC=?
{(0, 1, 0), (0, 1, 1), …, (1, 2, 2)}
Students  Courses
|Students| = m
|Courses| = n
|Students  Courses| = ?
Exercises
Show that:
A  (B  C) = (C  B)  A
A  (A  B) = A
A  (A  B) = A
Venn Diagram:
A  (B – C)
(A  B)  (A  C)
Latihan Soal
&
Pembagian Kelompok