Definisi Grup dan Contohnya
Summary
TLDRThis video script discusses the concept of groups in algebraic structures, a fundamental topic in abstract algebra. It defines a group as a set that satisfies four conditions: closure, associativity, identity, and invertibility. Examples of groups include the set of integers under addition and the set of integers modulo 4 under addition. The script also provides a non-example of a group, illustrating the importance of each condition. It invites viewers to engage with questions in the comments section and looks forward to future discussions.
Takeaways
- 🔢 A group in algebraic structures is defined by four criteria: closure, associativity, identity, and invertibility.
- 🧩 Closure means that for any two elements 'a' and 'b' in the group, their combination (like 'a*b') must also be in the group.
- 🔗 Associativity ensures that for any elements 'a', 'b', and 'c', the equation (a*b)*c = a*(b*c) holds true.
- 🌀 Identity refers to the presence of an element that, when combined with any group element, leaves that element unchanged.
- ↩️ Invertibility states that every element in the group has an inverse, such that combining the element with its inverse results in the identity element.
- 🔄 The set of integers (Z) under the operation of addition is an example of a group, satisfying all four group criteria.
- 🔄 The set Z4 under the operation of addition modulo 4 also forms a group, demonstrating closure, associativity, identity (0), and invertibility.
- 🚫 The set of integers under the operation of subtraction does not form a group because it fails the associativity criterion.
- 📚 Examples of other groups include the set of real numbers (R) under addition, and the set of non-zero real numbers under multiplication.
- ❌ The set of integers under multiplication does not form a group because not all elements have inverses (e.g., the inverse of 2 is 1/2, which is not an integer).
Q & A
What is a group in the context of algebraic structures?
-A group is one of the fundamental concepts in algebraic structures, defined as a set with an operation that satisfies four conditions: closure, associativity, identity, and invertibility.
What does it mean for a group to be 'closed'?
-A group is 'closed' if for any two elements a and b in the group, the result of their operation (a * b) is also in the group.
What is the associative property in the context of groups?
-The associative property means that for any elements a, b, and c in the group, the equation (a * b) * c = a * (b * c) holds true.
What is the identity element in a group?
-The identity element in a group is an element that, when combined with any other element in the group using the group operation, leaves that element unchanged.
What does it mean for every element in a group to have an inverse?
-Every element in a group has an inverse if for every element a in the group, there exists an element b such that a * b = b * a = identity element.
Can you provide an example of a group from the script?
-Yes, the set of integers (Z) under the operation of addition is given as an example of a group in the script.
Why is the set of integers modulo 4 (Z4) under addition considered a group?
-Z4 under addition is considered a group because it satisfies all four group properties: closure, associativity, identity (0 is the identity element), and every element has an inverse.
What is an example of a set that is not a group as per the script?
-The set of integers under the operation of subtraction is given as an example of a non-group because it does not satisfy the associative property.
What are some other examples of groups mentioned in the script?
-Other examples of groups mentioned include the set of real numbers (R) under addition, the set of non-zero real numbers under multiplication, and the set of 2x2 matrices with integer entries under matrix addition.
Why is the set of integers under multiplication not considered a group?
-The set of integers under multiplication is not considered a group because not every element has an inverse (for example, the integer 2 does not have an integer multiplicative inverse).
What is the significance of understanding groups in algebraic structures?
-Understanding groups in algebraic structures is significant because groups are fundamental in studying symmetries and transformations, which are essential in various areas of mathematics and its applications.
Outlines
📚 Introduction to Groups in Algebra
The speaker begins by introducing the concept of a group in algebra, a fundamental topic in the study of algebraic structures. They define a group as a set that satisfies four conditions: closure, associativity, identity, and invertibility. The explanation includes the closure property, which means that for any two elements a and b in the group, the result of their operation (denoted by *) must also be in the group. Associativity is demonstrated through the operation of elements a, b, and c, where a * (b * c) equals (a * b) * c. The identity element is introduced as an element that, when combined with any other element in the group through the operation, leaves that element unchanged. Lastly, the invertibility condition is explained, stating that for every element in the group, there must be an inverse element such that their operation results in the identity element. The integers under addition (denoted by Z) are given as an example of a group that meets all these criteria.
🔍 Examples and Non-Examples of Groups
In this section, the speaker provides further examples of groups, including the set of real numbers (R) under addition, the set of non-zero real numbers under multiplication, and the set of 2x2 matrices with integer entries under matrix addition. They also present a counterexample to illustrate a set that does not form a group. Specifically, they discuss the set of integers (Z) under subtraction, explaining why it fails to meet the criteria for a group. The closure property is satisfied, but the operation of subtraction is not associative, which is demonstrated through a simple arithmetic example. The speaker concludes by inviting viewers to ask questions about groups or algebraic structures in the comment section and looks forward to the next video.
Mindmap
Keywords
💡Group
💡Closure
💡Associativity
💡Identity
💡Invertibility
💡Integers (Z)
💡Modular Addition
💡Non-Group
💡Real Numbers (R)
💡Matrix
Highlights
Definition of a group in algebraic structures
A group is denoted by an asterisk and must satisfy four conditions
Closure property in groups
Associative property within groups
Identity element requirement for groups
Existence of inverse elements in a group
Example of a group: Integers under addition
Integers under addition satisfy the closure property
Associativity of addition for integers
Zero as the identity element in integer addition
Every integer has an additive inverse
Example of a group: Integers modulo 4 under addition
Closure, associativity, and identity in integers modulo 4
Inverses in integers modulo 4 addition
Real numbers under addition as a group
Non-zero real numbers under multiplication as a group
2x2 matrices with integer entries under matrix addition as a group
Integers under subtraction do not form a group due to non-associativity
Integers under multiplication do not form a group due to the lack of multiplicative inverses for all elements
Invitation for questions on groups and algebraic structures in the comments
Transcripts
kau tak tahu salam matematika Pada
kesempatan kali ini saya akan membahas
tentang definisi grup dan
contoh-contohnya pada matakuliah
struktur aljabar ingin lengkapnya simak
video berikut ini oke group adalah salah
satu materi dasar dalam mempelajari
matakuliah struktur aljabar kita dia
definisi suatu grup yang disimbolkan
oleh bintang adalah sebuah bintang ia
memenuhi empat syarat berikut yang
pertama adalah tertutup yang kedua
asosiatif yang ketiga identitas dan yang
keempat invers
Hai sahabat yang pertama yakni tertutup
maksudnya adalah setiap elemen adige dan
beige maka akan berakibat a jika
dioperasikan dengan b akan berada di
syarat yang berikutnya yakni asosiatif
maksudnya adalah setiap elemen
a-b-c-d-e-f-g-a maka akan berlaku a
bintang b bintang c = a bintang b
Simpang C syarat yang ketika ia
identitas artinya adalah sebuah grup
harus memiliki cepat berbuah
elemen-elemen disini maksudnya adalah
setiap kita ambil elemen adige maka akan
berakibat a bintang oke sama dengan aa3w
bintang A1 dengan A300 yang keempat
yakni invest maksudnya adalah setiap
elemen
elemen identitas saya tampilkan sebuah
contoh dari grup contohnya adalah
himpunan bilangan bulat Z terhadap
operasi penambahan buktinya
Hai yang pertama adalah zat tertutup
terhadap operasi penambahan karena
setiap kita ambil elemen zb6 NZ ditambah
B pastilah Element Z atau bilangan bulat
yang kedua operasi tambah kita sudah
tahu akan berlaku asosiatif pada
himpunan bilangan bulat untuk sehat yang
ketiga
Hai terdapat sebuah elemen identitas
pada operasi tambah pada himpunan
bilangan bulat tentunya yaitu nol
maksudnya adalah setiap elemen pada
bilangan bulat jika ditambahkan dengan
nol maka akan menghasilkan bilangan itu
sendiri dan untuk yang secara terakhir
setiap elemen pada himpunan bilangan
bulat operasi tambah pasti milikin Bos
setiap kita ambil Wah maka akan selalu
ada negatif a-yong merupakan ciri khas
dari a untuk contoh berikutnya
Hai hiburan zenpad 0123 terhadap operasi
penambahan modul 4 akan kita tunjukkan
bahwasanya ini merupakan contoh dari
grup Perhatikan tabel berikut Apakah
tertutup Iya berapa Iya karena setiap
hasil dari menambahkan Z4 dengan operasi
modul 4 akan menghasilkan elemen-elemen
yang juga berada di setempat Apakah
asosiatif siapanya Iya karena operasi
tambah pada himpunan bilangan bulat
mobil Oke pasti bersifat asosiatif yang
berikutnya Apakah milik identitas
jawabannya iya perhatikan 00 merupakan
elemen dari Z4 sedangkan semua
bilangan-bilangan pada tempat jika
ditambahkan
10 akan menghasilkan bilangan itu
sendiri lalu ditambah 001 kita merasa
berhak satu dan lain-lainnya kemudian
Apakah setiap elemen memiliki invers
jawabannya adalah Iya lepas dari 0013
karena 13022 karena dua tambah dua
adalah roh dari setiap elemen memiliki
invers ada banyak sekali contoh-contoh
dari grup antara lain himpunan bilangan
real R terhadap operasi penambahan
kemudian himpunan bilangan riil er tak
nol terhadap operasi perkalian atau bisa
juga matriks himpunan matriks 2 * dua
dengan entry bilangan bulat terhadap
operasi penambahan matriks dan
sebagainya
Hai berikut akan saya tampilkan sebuah
contoh himpunan dengan operasi biner
tertentu yang bukan merupakan grup
Hai contohnya adalah himpunan bilangan
bulat set terhadap operasi pengurangan
Mengapa bukan Bro Apakah tertutup
siapanya Iya kita tahu bahwasanya
bilangan bulat jika dikurangi bilangan
bulat hasilnya tetap berada di bilangan
bulat yang berikutnya apa asosiatif
jawabanya tidak ambil sebuah kontes yang
paling 54 dan 35 dikurangin Pak kemudian
dikurangi tiga tidak sama hasilnya
dengan lima dikurangi tempat tidurnya
tiga dari sana dapat dikatakan operasi
biner pada bulan tersebut tidak berlaku
asosiatif sehingga contoh berikut
bukanlah sebuah grup contoh bukan grup
berikutnya himpunan bilangan bulat dan
terhadap operasi perkalian kita cek satu
persatu
akan tertutup jawabannya Iya kenapa ya
Coba kalikan bilangan bulat dengan
bilangan gula pasti hasilnya selalu
bilangan bulat yang kedua asosiatif kita
sudah tahu bahwasanya perkalian pada
bilangan bulat berlaku sifat asosiatif
yang ketiga apakah memiliki elemen
identitas jawabannya Iya apa itu satu
semua bilangan bulat jika dikalikan
dengan satu akan menghasilkan bilangan
bulat tersendiri untuk yang keempat
syarat infus yang tidak terpenuhi kenapa
ambil kontras simple yakni 22 merupakan
bilangan bulat invers perkalian pada
bilangan bulat dua adalah setengah kita
tahu setengah bukanlah bilangan bulat
dari sana tidak terpenuhi syarat grup
sehingga contoh berikut bukanlah flu
Hai kamu Oke terima kasih sudah
menyaksikan video ini sampai tuntas jika
ada pertanyaan tentang grup maupun
tentang mata kuliah struktur aljabar
silakan tulis di kolom komentar berikut
ini sampai ketemu di video berikutnya
salam Atika
تصفح المزيد من مقاطع الفيديو ذات الصلة
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