The document defines sets and provides examples of sets. It discusses:
- The definition of a set as a well-defined collection of objects or elements.
- Examples of concrete and abstract sets.
- Notation used for sets such as curly brackets and membership using the epsilon symbol.
- The empty set, which contains no elements.
- Operations on sets such as unions, intersections, complements, differences, and Cartesian products.
Dokumen tersebut merupakan lembar kegiatan peserta didik tentang penyajian data dalam bentuk tabel distribusi frekuensi dan histogram. Siswa diajak mengingat macam-macam tabel distribusi, mengerjakan soal untuk disajikan dalam histogram, tabel distribusi frekuensi kumulatif dan ogive, serta mengerjakan soal secara kelompok dan individu.
Dokumen tersebut berisi soal-soal ujian tentang konsep-konsep geometri bidang datar khususnya lingkaran, busur lingkaran, dan segitiga. Di antaranya adalah menentukan luas, keliling, panjang busur lingkaran, besar sudut, dan hubungan antara bagian-bagian lingkaran berdasarkan informasi yang diberikan.
Dokumen tersebut merupakan lembar kegiatan peserta didik tentang penyajian data dalam bentuk tabel distribusi frekuensi dan histogram. Siswa diajak mengingat macam-macam tabel distribusi, mengerjakan soal untuk disajikan dalam histogram, tabel distribusi frekuensi kumulatif dan ogive, serta mengerjakan soal secara kelompok dan individu.
Dokumen tersebut berisi soal-soal ujian tentang konsep-konsep geometri bidang datar khususnya lingkaran, busur lingkaran, dan segitiga. Di antaranya adalah menentukan luas, keliling, panjang busur lingkaran, besar sudut, dan hubungan antara bagian-bagian lingkaran berdasarkan informasi yang diberikan.
1. Filsafat Yunani kuno Democritus dan Leucipus pertama kali mengemukakan gagasan tentang atom sebagai partikel terkecil materi pada abad ke-4 SM.
2. Atom adalah partikel terkecil suatu unsur yang tidak dapat diuraikan lagi melalui reaksi kimia.
3. Partikel subatom seperti proton dan elektron dapat memiliki muatan listrik positif atau negatif.
LATIHAN SOAL FISIKA SEMESTER GENAP KELAS 8qiera.id
Teks tersebut berisi soal-soal tentang gelombang dan optika yang mencakup konsep-konsep seperti frekuensi, panjang gelombang, pemantulan cahaya, lensa, cermin, dan pembentukan bayangan. Soal-soal tersebut meminta pembaca untuk memahami konsep-konsep tersebut dan memilih jawaban yang tepat berdasarkan gambar atau data yang diberikan.
1. Vektor adalah kuantitas fisik yang membutuhkan informasi tentang besarnya dan arahnya. Vektor dapat direpresentasikan secara geometris dengan panah yang panjangnya mewakili besar dan arah panah mewakili arah vektor.
2. Terdapat beberapa operasi pada vektor seperti penjumlahan, pengurangan, perkalian skalar, dan produk skalar dan silang. Penjumlahan vektor dapat dilakukan dengan metode jajaran
Dokumen tersebut membahas tentang pangkat rasional, bentuk akar, dan persamaan bentuk pangkat. Pertama, dijelaskan definisi dan sifat-sifat operasi pangkat seperti pangkat bulat positif, negatif, dan nol. Kedua, dijelaskan definisi dan operasi aljabar bentuk akar. Ketiga, diberikan contoh-contoh soal dan penyelesaian persamaan yang melibatkan bentuk pangkat dan akar.
Soal Tes Formatif yang dapat digunakan oleh guru Fisika di SMP untuk mengukur sampai sejauh mana pemahaman Konsep pada bab yang dimaksud telah tercapai. Soal ini dapat Anda gunakan sebagai Ulangan harian per KD yang dibahas pada masing-masing kelas. Semoga bermamfaat! Untuk lebih detailnay kunjungi saya pada http://aguspurnomosite.blogspot.com/
Dokumen tersebut merangkum tentang:
1. Induksi matematika sebagai teknik pembuktian untuk pernyataan perihal bilangan bulat
2. Prinsip kerja induksi matematika yaitu langkah basis dan langkah induksi
3. Contoh soal pembuktian dengan induksi matematika
Rencana pelaksanaan pembelajaran mata pelajaran matematika kelas XI semester ganjil SMK Negeri 2 Doloksanggul tahun pelajaran 2012-2013 membahas tentang perbandingan trigonometri, meliputi pengertian, rumus, dan penerapannya dalam menentukan unsur-unsur segitiga siku-siku dan masalah-masalah terkait lainnya. Materi akan disampaikan menggunakan metode ceramah, diskusi, penugasan, dan penemuan, di
Kartu soal merangkum 15 soal ujian kenaikan kelas mata pelajaran IPA tentang listrik statis dan dinamis. Soal-soal terdiri dari pilihan ganda dan uraian mengenai konsep medan listrik, hukum Ohm, energi listrik, dan proses terjadinya muatan pada benda.
Dokumen menjelaskan operasi aljabar pada fungsi, termasuk penjumlahan, pengurangan, perkalian, dan pembagian fungsi. Contoh soal ditanyakan untuk menemukan hasil operasi aljabar dari dua fungsi f(x) dan g(x) yang diberikan.
The document provides information about sets including:
1) Sets can be finite, infinite, empty, or singleton and are represented using curly brackets. Common set operations are union, intersection, and complement.
2) A set's cardinality refers to the number of elements it contains. Subsets are sets contained within other sets.
3) Examples are given of representing sets using roster and set-builder methods and performing set operations like union, intersection, and complement on sample sets.
4) Basic properties of sets like idempotent, commutative, and associative laws for simplifying set expressions are outlined.
The document defines basic set theory concepts including:
- A set is a collection of distinct objects called elements.
- Sets can be represented using curly brackets or set builder notation.
- The empty set, subsets, proper subsets, unions, intersections and complements are defined.
- Cardinality refers to the number of elements in a finite set.
- Power sets are the set of all subsets of a given set.
1. Filsafat Yunani kuno Democritus dan Leucipus pertama kali mengemukakan gagasan tentang atom sebagai partikel terkecil materi pada abad ke-4 SM.
2. Atom adalah partikel terkecil suatu unsur yang tidak dapat diuraikan lagi melalui reaksi kimia.
3. Partikel subatom seperti proton dan elektron dapat memiliki muatan listrik positif atau negatif.
LATIHAN SOAL FISIKA SEMESTER GENAP KELAS 8qiera.id
Teks tersebut berisi soal-soal tentang gelombang dan optika yang mencakup konsep-konsep seperti frekuensi, panjang gelombang, pemantulan cahaya, lensa, cermin, dan pembentukan bayangan. Soal-soal tersebut meminta pembaca untuk memahami konsep-konsep tersebut dan memilih jawaban yang tepat berdasarkan gambar atau data yang diberikan.
1. Vektor adalah kuantitas fisik yang membutuhkan informasi tentang besarnya dan arahnya. Vektor dapat direpresentasikan secara geometris dengan panah yang panjangnya mewakili besar dan arah panah mewakili arah vektor.
2. Terdapat beberapa operasi pada vektor seperti penjumlahan, pengurangan, perkalian skalar, dan produk skalar dan silang. Penjumlahan vektor dapat dilakukan dengan metode jajaran
Dokumen tersebut membahas tentang pangkat rasional, bentuk akar, dan persamaan bentuk pangkat. Pertama, dijelaskan definisi dan sifat-sifat operasi pangkat seperti pangkat bulat positif, negatif, dan nol. Kedua, dijelaskan definisi dan operasi aljabar bentuk akar. Ketiga, diberikan contoh-contoh soal dan penyelesaian persamaan yang melibatkan bentuk pangkat dan akar.
Soal Tes Formatif yang dapat digunakan oleh guru Fisika di SMP untuk mengukur sampai sejauh mana pemahaman Konsep pada bab yang dimaksud telah tercapai. Soal ini dapat Anda gunakan sebagai Ulangan harian per KD yang dibahas pada masing-masing kelas. Semoga bermamfaat! Untuk lebih detailnay kunjungi saya pada http://aguspurnomosite.blogspot.com/
Dokumen tersebut merangkum tentang:
1. Induksi matematika sebagai teknik pembuktian untuk pernyataan perihal bilangan bulat
2. Prinsip kerja induksi matematika yaitu langkah basis dan langkah induksi
3. Contoh soal pembuktian dengan induksi matematika
Rencana pelaksanaan pembelajaran mata pelajaran matematika kelas XI semester ganjil SMK Negeri 2 Doloksanggul tahun pelajaran 2012-2013 membahas tentang perbandingan trigonometri, meliputi pengertian, rumus, dan penerapannya dalam menentukan unsur-unsur segitiga siku-siku dan masalah-masalah terkait lainnya. Materi akan disampaikan menggunakan metode ceramah, diskusi, penugasan, dan penemuan, di
Kartu soal merangkum 15 soal ujian kenaikan kelas mata pelajaran IPA tentang listrik statis dan dinamis. Soal-soal terdiri dari pilihan ganda dan uraian mengenai konsep medan listrik, hukum Ohm, energi listrik, dan proses terjadinya muatan pada benda.
Dokumen menjelaskan operasi aljabar pada fungsi, termasuk penjumlahan, pengurangan, perkalian, dan pembagian fungsi. Contoh soal ditanyakan untuk menemukan hasil operasi aljabar dari dua fungsi f(x) dan g(x) yang diberikan.
The document provides information about sets including:
1) Sets can be finite, infinite, empty, or singleton and are represented using curly brackets. Common set operations are union, intersection, and complement.
2) A set's cardinality refers to the number of elements it contains. Subsets are sets contained within other sets.
3) Examples are given of representing sets using roster and set-builder methods and performing set operations like union, intersection, and complement on sample sets.
4) Basic properties of sets like idempotent, commutative, and associative laws for simplifying set expressions are outlined.
The document defines basic set theory concepts including:
- A set is a collection of distinct objects called elements.
- Sets can be represented using curly brackets or set builder notation.
- The empty set, subsets, proper subsets, unions, intersections and complements are defined.
- Cardinality refers to the number of elements in a finite set.
- Power sets are the set of all subsets of a given set.
1) The document introduces fundamental concepts of set theory including sets, elements, subsets, cardinality, power sets, Cartesian products, and set operations like union, intersection, difference, and complement.
2) Key concepts are explained through examples such as defining rational and real number sets. Properties of subsets, cardinality of power sets, and Cartesian products of sets are covered.
3) Set operations like union, intersection, difference and complement are defined along with proofs of algebraic properties like distributivity through truth tables. Exercises are suggested to practice converting between logical and set expressions.
The document discusses logical equivalence and provides a truth table to show that the statements (P ∨ Q) and (¬P) ∧ (¬Q) are logically equivalent. It shows that ¬(P ∨ Q) ↔ (¬P) ∧ (¬Q) is true according to the truth table.
This document provides an overview of basic set theory concepts including defining and representing sets, the number of elements in a set, comparing sets, subsets, and operations on sets including union and intersection. Key points covered are defining a set as a collection of well-defined objects, representing sets using listing, defining properties, or Venn diagrams, defining the cardinal number of a set as the number of elements it contains, comparing sets as equal or equivalent based on elements, defining subsets as sets contained within other sets, and defining union as the set of elements in either set and intersection as the set of elements common to both sets.
The document provides information about sets, relations, and functions in mathematics:
- A set is a collection of distinct objects, called elements or members. Sets are represented using curly brackets and elements are separated by commas. There are finite and infinite sets. Operations on sets include union, intersection, complement, difference, and power set.
- A relation from a set A to a set B is a subset of the Cartesian product A × B. The domain is the set of first elements in the relation and the range is the set of second elements.
- A function from a set A to a set B is a special type of relation where each element of A is mapped to exactly one element of B.
This document provides an overview of basic discrete mathematical structures including sets, functions, sequences, sums, and matrices. It begins by defining a set as an unordered collection of elements and describes various ways to represent sets, such as listing elements or using set-builder notation. It then discusses operations on sets like unions, intersections, complements, and Cartesian products. Finally, it introduces functions as assignments of elements from one set to another. The document serves as an introduction to fundamental discrete structures used throughout mathematics.
This document defines and explains key concepts in sets and set theory, including:
- Defining sets and listing elements using roster and set-builder notation
- Important mathematical sets like natural numbers, integers, and real numbers
- Describing relationships between sets such as subsets, equality, and Venn diagrams
- Calculating cardinality to determine the number of elements in a set
- Forming Cartesian products to combine elements from multiple sets into ordered pairs
The document defines and describes sets. Some key points include:
- A set is a collection of well-defined objects. Sets can be described in roster form or set-builder form.
- There are different types of sets such as the empty/null set, singleton sets, finite sets, and infinite sets.
- Set operations include union, intersection, difference, symmetric difference, and complement. Properties like subsets and the power set are also discussed.
- Cardinality refers to the number of elements in a set. Formulas are given for finding the cardinality of sets under different operations.
This document discusses sets, functions, and relations. It begins by defining key terms related to sets such as elements, subsets, operations on sets using union, intersection, difference and complement. It then discusses relations and functions, defining a relation as a set of ordered pairs where the elements are related based on a given condition. Functions are introduced as a special type of relation where each element of the domain has a single element in the range. The document aims to help students understand and use the properties and notations of sets, functions, and relations, and appreciate their importance in real-life situations.
A set is an unordered collection of objects called elements. A set contains its elements. Sets can be finite or infinite. Common ways to describe a set include listing elements between curly braces or using a set builder notation. The cardinality of a set refers to the number of elements it contains. Power sets contain all possible subsets of a given set. Cartesian products combine elements of sets into ordered pairs. Common set operations include union, intersection, difference, and testing if sets are disjoint.
The document discusses sets and functions. It introduces basic set theory concepts like sets, elements, subsets, set operations, and functions. It defines what a set is, provides examples of common sets like natural numbers and integers, and covers topics like set equality, subsets, cardinality, and the power set. It also defines what a function is, including domain, codomain, range, and using an example to illustrate a function that maps people to cities.
The document provides information about sets, relations, and functions. It defines what a set is and how sets can be described in roster form or set-builder form. It discusses different types of sets such as the empty set, singleton sets, finite sets, and infinite sets. It also covers topics like subsets, power sets, operations on sets such as union, intersection, difference, complement, and applications of these concepts to solve problems involving cardinality of sets.
The document provides information about sets including definitions of key terms like union, intersection, complement, difference, properties of these operations, and counting theorems. It discusses describing sets by explicitly listing members or through a relationship. Examples are provided to illustrate concepts like subsets, proper subsets, power sets, De Morgan's laws, and using Venn diagrams to solve problems involving sets. Counting theorems are presented to calculate the number of elements in unions, intersections, and complements of finite sets.
This document provides notes on discrete mathematics. It begins by defining the two main types of mathematics: continuous mathematics, which involves real numbers and smooth curves, and discrete mathematics, which involves distinct, countable values between points.
The document then lists some common topics in discrete mathematics, such as sets, logic, graphs, and trees. It provides an overview of sets and set theory, including defining sets, representing sets, membership, important sets like natural and real numbers, and describing properties like cardinality, finite vs infinite sets, subsets, and empty/singleton/equal/equivalent sets. It also introduces basic set operations like union, intersection, difference, and Cartesian product.
This document provides an overview of set theory concepts including:
- Sets, elements, and set operations like union, intersection, difference, and complement.
- Finite and countable sets versus infinite sets.
- Product sets involving ordered pairs from two sets.
- Classes of sets including the power set of a set, which contains all subsets.
- A set is a well-defined collection of objects. The empty set is a set with no elements. A set is finite if it has a definite number of elements, otherwise it is infinite.
- Two sets are equal if they have exactly the same elements. A set A is a subset of set B if every element of A is also an element of B.
- The power set of a set A is the collection of all subsets of A, including the empty set and A itself. It is denoted by P(A).
About sets , definition example, and some types of set. Explained the some operation of set like union of set and intersection of set with usual number example
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
2. Definisi
Himpunan
Himpunan adalah kumpulan benda-benda dan unsur-unsur
yang didefinisikan dengan jelas dan juga diberi batasan
tertentu
Dalam pengertian yang lebih lengkap, himpunan adalah
kumpulan suatu benda baik kongkrit (nyata) ataupun
abstrak yang berada dalam suatu tempat sesuai dengan
sifat tertentu,
Benda kongkrit ataupun abstrak yang terdapat dalam
himpunan disebut elemen atau anggota himpunan,
biasanya ditulis di antara dua kurung kurawal notasi ϵ.
Sedangkan,himpunan yang tidak mempunyai anggota
disebut himpunan kosong. Nama himpunan biasanya
dinotasikan dengan huruf kapital.
Contoh, benda a menjadi anggota himpunan K dapat
dinyatan dengan a ϵ K. Sedangkan, banyaknya anggota
himpunan K yang berhingga dinotasikan dengan n (K)
3. Contoh Himpunan
perhatikan contoh kasus berikut ini!
a) Kumpulan pemuda ganteng
b) Kumpulan orang tua yang bijaksana
c) Kumpulan pena, buku, penggaris, penghapus, pensil
d) Kumpulan pisang, salak, duku, durian, rambutan, jeruk
Penjelasan
Pada contoh (a) kumpulan pemuda ganteng; pengertian
ganteng itu relatif dan tidak dapat didefinisikan dengan
jelas, dan (b) sifat bijaksana juga merupakan hal yang
tidak dapat didefinisikan dengan jelas karena setiap orang
memiliki penilaian yang berbeda-beda (relatif).
3
4. 4
Cara Menyajikan
Himpunan
1. Menyatakan himpunan dengan menggunakan kata-kata atau menyebut
syarat-syaratnya
Conyohnya adalah;
A = { bilangan prima kurang dari 20 }
B = { bilangan asli antara 7 sampai 25 }
2. Menyatakan himpunan dengan menyebutkan atau mendaftar anggota
anggotanya
Yaitu dengan cara anggota himpunan dituliskan di dalam kurung kurawal dan antara
anggota yang satu dengan yang lainnya dipisahkan dengan tanda koma.
Contohnya adalah;
A = { jeruk, salak, jambu, semangka, mangga }
(untuk himpunan yang anggotanya sedikit atau terbatas)
B = { Aceh, Medan, Padang, Palembang, Bengkulu, Lampung, … Makasar }
(untuk himpunan yang anggotanya banyak tapi terbatas)
C = { 2, 3, 5, 7, 11, 13, ..... }
(untuk himpunan yang jumlah anggotanya banyak dan tidak terbatas)
5. 3. Menyatakan himpunan dengan notasi pembentuk
himpunan
Cara menyatakan himpunan dengan notasi pembentuk
himpunan adalah dengan mengikuti aturan berikut ini;
a) Benda atau objeknya dilambangkan dengan sebuah
peubah (a, b, c, ...., z)
b) Menuliskan syarat keanggotaannya dibelakang tanda
‘I’
Contohnya adalah;
A = { x I x < 7, x bilangan asli }
Dibaca: Untuk x anggota himpunan A dimana x kurang
dari 7 dan x adalah bilangan asli.
B = { (x,y) I y + x = 7, x dan y bilangan asli }
Dibaca: himpunan pasangan x dan y sedemikian
sehingga y ditambah x sama dengan 7 untuk x dan y
adalah bilangan asli.
5
6. 4. Menyatakan Himpunan Dengan Diagram Venn
Contoh
Misalkan U = {1, 2, …, 7, 8},
A = {1, 2, 3, 5} dan B = {2, 5, 6, 8}
6
7. Jumlah elemen di dalam A disebut kardinal dari
himpunan A.
Notasi: n(A) atau | A |
Contoh
(i) B = { x | x merupakan bilangan prima lebih kecil dari
20 },
B = {2, 3, 5, 7, 11, 13, 17, 19} maka | B | = 8
(ii) T = {kucing, a, Amir, 10, paku}, maka | T | = 5
(iii) A = {a, {a}, {{a}} }, maka | A | = 3
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Kardinalitas
8. Himpunan Kosong (Null
set)
Himpunan dengan kardinal = 0 disebut himpunan kosong
(null set).
Notasi : atau {}
Contoh
(i) E = { x | x < x }, maka n(E) = 0
(ii) P = { orang Indonesia yang pernah ke bulan }, maka n(P) = 0
(iii) A = {x | x adalah akar persamaan kuadrat 𝑋2 + 1 = 0 },
n(A) = 0
himpunan {{ }} dapat juga ditulis sebagai {}
himpunan {{ }, {{ }}} dapat juga ditulis sebagai {, {}}
{} bukan himpunan kosong karena ia memuat satu elemen yaitu
himpunan kosong
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9. Himpunan Bagian
(Subset)
▪ Himpunan A dikatakan himpunan bagian dari himpunan
B jika dan hanya jika setiap elemen A merupakan
elemen dari B.
▪ Dalam hal ini, B dikatakan superset dari A.
▪ Notasi: A B
▪ Diagram Venn
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10. Contoh
(i) { 1, 2, 3} {1, 2, 3, 4, 5}
(ii) {1, 2, 3} {1, 2, 3}
(iii) Jika A = { (x, y) | x + y < 4, (x , y) ≥ 0 } dan
B = { (x, y) | 2x + y < 4, x ≥ 0 dan y ≥ 0 }, maka B A.
TEOREMA 1. Untuk sembarang himpunan A berlaku hal-hal
sebagai berikut:
(a) A adalah himpunan bagian dari A itu sendiri (yaitu, A A).
(b) Himpunan kosong merupakan himpunan bagian dari A
( A).
(c) Jika A B dan B C, maka A C
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11. Himpunan Yang
Sama
▪ A = B jika dan hanya jika setiap elemen A merupakan
elemen B dan sebaliknya setiap elemen B merupakan
elemen A.
▪ A = B jika A adalah himpunan bagian dari B dan B
adalah himpunan bagian dari A. Jika tidak demikian,
maka A B.
▪ Notasi : A = B A B dan B A
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12. Contoh
(i) Jika A = { 0, 1 } dan B = { x | x (x – 1) = 0 }, maka A = B
(ii) Jika A = { 3, 5, 8, 5 } dan B = {3, 8}, maka A ≠ B
Untuk tiga buah himpunan, A, B, dan C berlaku aksioma
berikut:
(a) A = A, B = B, dan C = C
(b) jika A = B, maka B = A
(c) jika A = B dan B = C, maka A = C
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13. Himpunan Yang
Ekuivalen
▪ Himpunan A dikatakan ekuivalen dengan himpunan B
jika dan hanya jika kardinal dari kedua himpunan
tersebut sama.
▪ Notasi : A ~ B A = B
▪ Contoh
Misalkan A = { 1, 3, 5, 7 } dan B = { a, b, c, d }, maka
A ~ B sebab A = B = 4
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14. Himpunan Saling
Lepas
▪ Dua himpunan A dan B dikatakan saling lepas (disjoint)
jika keduanya tidak memiliki elemen yang sama.
▪ Notasi : A // B
▪ Diagram Venn:
▪ Contoh
Jika A = { x | x P, x < 8 } dan B = { 10, 20, 30, ... },
maka A // B
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15. Himpunan Kuasa
▪ Himpunan kuasa (power set) dari himpunan A adalah
suatu himpunan yang elemennya merupakan semua
himpunan bagian dari A, termasuk himpunan kosong dan
himpunan A sendiri.
▪ Notasi : P(A) atau 2A
▪ Jika A = m, maka P(A) = 2m.
Contoh 12.
Jika A = { 1, 2 }, maka P(A) = { , { 1 }, { 2 }, { 1, 2 }}
Contoh 13.
Himpunan kuasa dari himpunan kosong adalah P() =
{}, dan
himpunan kuasa dari himpunan {} adalah P({}) =
{, {}}
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16. Operasi Terhadap
Himpunan
▪ Irisan (intersection)
Notasi : A B = { x x A dan x B }
▪ Contoh
(i) Jika A = {2, 4, 6, 8, 10} dan B = {4, 10, 14, 18},
maka A B = {4, 10}
(ii) Jika A = { 3, 5, 9 } dan B = { -2, 6 }, maka A B
= . Artinya: A // B
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17. Gabungan
(Union)
▪ Notasi : A B = { x x A atau x B }
▪ Contoh
(i) Jika A = { 2, 5, 8 } dan B = { 7, 5, 22 }, maka
A B ={ 2, 5, 7, 8, 22 }
(ii) A = A
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18. Komplemen (complement)
▪ Notasi : 𝐴 = { x x U, x A }
▪ Contoh
Misalkan U = { 1, 2, 3, ..., 9 },
(i) jika A = {1, 3, 7, 9}, maka 𝐴 = {2, 4, 6, 8}
(ii) jika A = { x | x/2 P, x < 9 }, maka 𝐴 = { 1,
3, 5, 7, 9 }
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19. Selisih (difference)
▪ Notasi : A – B = { x x A dan x B } = A B
Contoh
(i) Jika A = { 1, 2, 3, ..., 10 } dan B = { 2, 4, 6, 8, 10 }
maka A – B = { 1, 3, 5, 7, 9 } dan B – A =
(ii) {1, 3, 5} – {1, 2, 3} = {5}, tetapi {1, 2, 3} – {1, 3, 5} = {2}
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20. Beda Setangkup (Symmetric Difference)
▪ Notasi: A B = (A B) – (A B) = (A – B) (B – A)
Contoh
Jika A = { 2, 4, 6 } dan B = { 2, 3, 5 },
maka A B = { 3, 4, 5, 6 }
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21. Perkalian Kartesian
(cartesian product)
▪ Notasi: A B = {(a, b) a A dan b B }
▪ Contoh
▪ Misalkan C = { 1, 2, 3 }, dan D = { a, b }, maka
C D = { (1, a), (1, b), (2, a), (2, b), (3, a), (3, b) }
▪ Misalkan A = B = himpunan semua bilangan riil, maka
A B = himpunan semua titik di bidang datar
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