2. Anggota Kelompok
1. Azzah Shaffiyah (2065061001)
2. Beltra Saura Rahmadan (2015061029)
3. Muhamad Ferdian Hidayat (2015061030)
4. Muhammad Rizky Rifaldi (2055061002)
5. Rahmad Romadhoni (2015061033)
6. Zaki Taufiqurrachman (2015061034)
3. • Sebuah bahasa dinyatakan
memiliki ekspresi regular jika
terdapat Finite State Automata
(FSA) yang dapat menerimanya.
Ekspresi Reguler
• Bahasa-bahasa yang diterima
oleh FSA bisa dinyatakan secara
sederhana dengan ekspresi
regular (regular expression).
4. bisa tidak muncul,
bisa juga muncul
berhingga kali
ER : ab*cc, 010*, a*d
berarti minimal
muncul satu kali
ER : a⁺d
union
ER : a * ∪b * , (a ∪b) *,01 * + 0
konkatenansi
Biasanya tanpa ditulis
titiknya, misal ab, berarti
sama dengan a.b
Notasi Reguler
* + + U
atau .(titik)
6. 01. Sifat Komutatif
dan Asosiatif
• Sifat Komutatif
Dapat membalik urutan operand-operand nya
dan tetap memperoleh hasil akhir yang sama.
Memungkinkan kita untuk mengelompokkan
operand nya Ketika operator dikenakan dua kali.
• Sifat Asosiatif
7. Sifat Komutatif & Asosiatif
yang berlaku pada ER
1) A + B = B + A
Hukum ini (hukum komutatif untuk gabungan) menyatakan
bahwa kita dapat melakukan gabungan dua bahasa
tersebut, baik dengan urutan seperti di sebelah kiri ‘=‘
maupun seperti di sebelah kanan ‘=‘
8. 2) (A + B ) + C = A + ( B + C )
Hukum ini (hukum asosiatif untuk gabungan) menyatakan bahwa kita dapat
melakukan gabungan pada tiga bahasa, baik dengan mengambil gabungan dua
bahasa pertama terlebih dahulu maupun dengan mengambil gabungan dau
bahasa terakhir.
3) ( AB ) C = A ( BC )
Hukum ini ( hukum asosiatif untuk penyambungan/concatenation) menyatakan
bahwa kita dapat merenteng tuga bahasa dengan menyambung dua bahasa
pertama terlebih dahulu atau dua bahasa terakhir terlebih dahulu.
9. Hukum : AB = BA tidak berlaku dalam ekspresi regular
Contoh : ekspresi regular 01 dan 10
Ekspresi tersebut berturut-turut melambangkan bahasa {01} dan {10}.
Ekspresi 0 untuk A dan 1 untuk B tidak dapat disubstitusi.
Karena bahasanya berbeda, aka hukum AB = BA tidak berlaku.
10. 02. Identidas dan Anihilator
a) Identitas suatu operator
Nilai yang sedemikian sehingga Ketika dikenakan pada identitas dan suatu
nilai lain, maka hasilnya nilai lain lagi.
Contoh :
0 adalah identitas untuk penjumlahan, karena
0 + ꭓ = ꭓ + 0 = ꭓ
1 adalah identitas untuk perkalian, karena
1 x ꭓ = ꭓ x 1 = ꭓ
11. b) Anihilator untuk suatu operator
Nilai yang sedemikian sehingga Ketika operator tersebut dikenakan pada
annihilator dan suatu nilai lain, hasilnya adalah annihilator.
Contoh :
0 adalah annihilator untuk perkalian, karena
0 x ꭓ = ꭓ x 0 = 0
12. Hukum identitas dan annihilator
yang berlaku pada ER
1) θ + L = L + θ = L
Hukum ini menegaskan bahwa θ adalah identitas untuk
operasi gabungan.
2) ϵ L = L ϵ = L
Hukum ini menegaskan bahwa ϵ adalah identitas untuk
operasi concatenation.
3) θ L = L θ = θ
Hukum ini menegaskan bahwa θ adalah annihilator untuk
operasi concatenation.
13. 03. Hukum Distributif
Hukum distributive melibatkan dua operator, dan menyatakan
bahwa salah satu operator dapat dipaksa untuk dikenakan pada
tiap-tiap argument operator lain secara individual.
Contoh :
Hukum distributive perkalian atas penjumlahan
ꭓ x ( y + z ) = ( ꭓ x y ) + ( ꭓ x z )
14. Hukum distributive yang
berlaku pada ER
1) A ( M + N ) = AM + AN
Hukum ini adalah hukum distributive kiri
concatenation terhadap gabungan (union).
2) ( M + N ) A = MA + NA
Hukum ini adalah hukum distributive kanan
concatenation terhadap gabungan (union).
Gabungan dan irisan adalah contoh sederhana
operator idempotent.
15. 04. Hukum Idempoten
Suatu Operator dikatakan idempotent jika hasil
penerapannya pada dua nilai yang sama sebagai
argument adalah nilai itu sendiri.
Operator aritmatika biasa tidak bersifat idempotent ;
ꭓ + ꭓ ≠ ꭓ dan ꭓ x ꭓ ≠ ꭓ
Gabungan dan irisan adalah contoh sederhana
operator idempotent.
16. Hukum idempotent yang
berlaku pada ER
1) L + L = L
Hukum ini (hukum idempoten untuk gabungan)
menyatakan bahwa jika kita mengambil
gabungan dari dua ekspresi yang identic, kita
dapat mengganti keduanya dengan satu Salinan
ekspresi tersebut.
17. Contoh Soal
1. Diberikan ekspresi regular 0 + 01*.
Penyelesaian :
Ekspresi tersebut dapat disederhanakan menggunakan hukum-hukum
aljabar dalam ekspresi regular:
0 + 01*
= 0ε + 01* dari (2b)
= 0(ε + 1*) dari (3a), distributif kiri
= 01* karena ε + R = R
18. Contoh Soal
2. (L + M )* = (L*M*)*
Penyelesaian :
Untuk menunjukkan kesamaan tersebut, ganti variabel L dan M berturut-turut
dengan symbol a dan b, sehingga diperoleh ekspresi regular (a+b)* dan (a*b*)*.
Kedua ekspresi regular tersebut menyatakan bahasa dengan semua string dari a
dan b.
Dengan demikian, kesamaan (L + M )* = (L*M*)* benar.
19. Contoh Soal
3. L* = L*L*
Penyelesaian :
Untuk menunjukkan kesamaan tersebut, ganti
variabel L dengan simbol a, sehingga diperoleh
ekspresi regular a* dan a*a*.
Kedua ekspresi regular tersebut menyatakan
bahasa dengan semua string dari a.
Dengan demikian, kesamaan L* =L*L* benar.
20. Contoh Soal
4. L + ML = (L + M)L
Penyelesaian :
Untuk menunjukkan kesamaan tersebut, ganti
variabel L dan M berturut-turut dengan simbol a
dan b, sehingga diperoleh ekspresi regular a+ba
dan (a+b)a.
Kedua ekspresi regular tersebut menyatakan
Bahasa yang berbeda.
Untuk menunjukkan hal tersebut, pilih aa dalam
bahasa dari ekspresi regular (a+b)a, tapi tidak
dalam bahasa dari ekspresi regular a+ba.
Dengan demikian, kesamaan L + ML = (L + M)L
salah
25. Mercury
It’s the smallest planet
of them all
Venus
Venus is the second
planet from the Sun
Jupiter
Jupiter is the biggest
planet of them all
Saturn
It’s composed of
hydrogen and helium
Mars
Mars is actually a very
cold place
Neptune
It’s the farthest planet
from the Sun
Algebraic expressions
26. Algebraic operations
Venus
Venus is the second
planet from the Sun
Jupiter
Jupiter is the biggest
planet of them all
Mars
Despite being red,
Mars is a cold place
Saturn
Saturn is a gas giant
and has several rings
27. A picture always
reinforces the
concept
Images reveal large amounts of data, so
remember: use an image instead of a long text.
Your audience will appreciate it
28. This is a map
Latin America
Venus has a beautiful name
and is the second planet
from the Sun
Europe
Despite being red, Mars is
actually a cold place. It’s full
of iron oxide dust
29. Steps to tackle equations
01
Read
problem
Venus is the
second planet
from the Sun
02
Identify
unknown
Jupiter is the
biggest planet of
them all
03
Approach
equation
Despite being
red, Mars is a
cold place
04
Solve
equation
Saturn is a gas
giant and has
several rings
05
Answer
question
Mercury is the
smallest planet of
them all
30. X - 5 = 3
Example of an equation
Unknown
Jupiter is the biggest
planet of them all
First member
Saturn is composed of
hydrogen and helium
Second member
Neptune is very far away
from Earth
31. Activities
● A number increased by 8 equals 22. What number is it? --
● Maria's age increased by 7 equals 15. How old is Maria? --
● Double a number equals 46. Find that number --
● If half of Marta's age is equal to 10, how old is Marta? --
● The triple of a number decreased by 1 is equal to 20. Find that number --
32. Do these exercises
Solve: Result
7 x 4 --
6 x 8 --
3 x 9 --
5 x 5 --
4 x 8 --
Solve: Result
3 x 6 --
2 x 8 --
9 x 2 --
7 x 9 --
2 x 6 --
33. Practical exercise
Exercise A
Venus has a beautiful
name and is the second
planet from the Sun
Exercise B
Despite being red, Mars
is actually a cold place.
It’s full of iron oxide dust
Result
--
Result
--
34. Interpretation exercise
Interpret a multiplication equation as a comparison, for example:
35 = 5 × 7 35 is 5 times as
many as 7
7 times as many
as 5
Comparison A
Despite being red, Mars is
a very cold place
Comparison B
Saturn is a gas giant and
has several rings
Interpret
Neptune is the farthest
planet from the Sun
35. Multiples
Observe and match:
Multiples of 5
Multiples of 9
Multiples of 6
Multiples of 8
Multiples of 7
Multiples of 4
12, 18, 24, 30
16, 24, 32, 40
15, 20, 25, 30
18, 27, 36,45
14, 28, 35, 42
28, 36, 40, 44
36. Important concepts
Factors
Jupiter is the biggest
planet of them all
Hexagons
Saturn is a gas giant
and has several rings
Equality
Neptune is very far
away from us
Logarithm
Ceres is located in
the asteroid belt
Binomial
Mercury is the
smallest planet
Cube
Venus is the second
planet from the Sun
Digit
Mars is actually a
very cold place
Exponent
Earth is the planet
where we live on
38. Tablet
app
You can replace the image on
the screen with your own work.
Just right-click on it and select
“Replace image”
39. Mobile
app
You can replace the image on the
screen with your own work. Just
right-click on it and select
“Replace image”
40. Susan Bones Timmy Jimmy
You can talk a bit about this
person here
You can talk a bit about this
person here
Our team
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