1. Adam AS
2. Idris As
3. Nuh AS
4. Hud AS
5. Shaleh AS
6. Ibrahim AS
7. Ismail AS
8. Luth AS
9. Ishaq AS
10. Ya'qub AS
11. Yusuf AS
12. Ayyub AS
13. Zulkifli AS
14. Syu'aib AS
15. Musa AS
16. Harun AS
17. Daud AS
18. Sulaiman AS
19. Ilyas AS
20. Ilyasa AS
21. Yunus AS
22. Zakaria AS
23. Yahya AS
24. Isa AS
25. Muhammad SAW
Sifat-sifat Allah
No Sifat Wajib Sifat Mustahil
1 Wujud Ada Adam Tidak ada
2 Qidam Dahulu Huduus Baru
3 Baqa' Kekal Fana Rusak
4 Mukhalafatuhu lil hawadits Berbeda dengan ciptaan-Nya Mumatsalatuhu lil hawadits Sama dengan ciptaan-Nya
5 Qiyamuhu binafsihi Berdiri dengan sendirinya Ihtiyaju lighairihi Membutuhkan yang lain
6 Wahdaniyyah Esa atau Tunggal Ta'addud Berbilang
7 Qudrah Berkuasa 'Ajzun Lemah
8 Iradah Berkehendak Karahah Terpaksa
9 Ilmu Mengetahui Jahlun Bodoh
10 Hayat Hidup Mautun Mati
11 Sam'un Mendengar Samamum Tuli
12 Basar Melihat Umyun Buta
13 Kalam Berkata Bukmun Bisu
14 Qadirun Yang Berkuasa 'Ajizun Yang maha lemah
15 Muridun Yang Berkehendak Mukrahun Yang maha terpaksa
16 'Alimun Yang Mengetahui Jahilun Yang maha bodoh
17 Hayyun Yang Hidup Mayyitun Yang mati
18 Sami'un Yang Mendengar Ashamma Yang maha tuli
19 Basirun Yang Melihat A'maa Yang maha buta
20 Mutakallimun Yang Berbicara Abkama Yang maha bisu
In an effort towards world-class universities (world class university) Indonesia need to know what actually is meant by the term, how to achieve it and whether it needs all Indonesian universities to pursue degree of this world-class university. That's Directorate General for Higher Education took the initiative to invite Professor Kai-Ming Cheng, professor of the University of Hong Kong to talk about this world-class university.
"I held talks with the World Bank, how to measure this world-class university, and who might that be brought in experts. World Bank says there are two people, namely Professor Jamil Salmi and Professor Kai-Ming Cheng. Finally we get professors Kai-Ming is to come face to face directly. SemenAtara with Professor Jamil Salmi, along with Minister of National Education before we have done teleconference for 3 hours "said Director General of Higher Education, Fasli Jalal, daa
[Professor Kai-Ming Cheng is a lecturer at the University of Hong Kong. If we look 100 Universtitas Top Brand in the world which was released in 2007 by the Times Higher Education Supplement, the University of Hong Kong are on the order of close to 18 major universities such as Harvard, Yale, Oxford, Cambridge, Massachusetts Institute of Technology. Professor Kai-Ming is a part of this success. He is one of the drafter of the idea to the World Class University at University of Hong Kong. He sat as a dean of science, educational, and never became deputy Rector of this sana.Saat Professor Kai-Ming filanropy is also director of Hong Kong responsible for Fund raising in the field of education. This institution was only 8 years old. Although still young, these institutions managed to collect money 100 million U.S. dollars per year. For the East Asia region is a fantastic amount. Therefore, half the time he was engaged as a professor and the other half as the director of this philanthropy.
In addition to that he was also a consultant to UNESCO, the World Bank and various institutions in the world for education, ranging from large countries to smaller countries. What was delivered by Professor is based on field experience, not just a purely theoretical reflections]
Prof. Kai-Ming gave a very interesting background why the importance of this world-class university. Currently, the Professor, the world is changing very rapidly, both in the social, organizational, job models, job activities, Front-Line Workers, Individual life and including expectations. Environmental scan was advanced by Professor Kai-Ming because higher education has been influenced by changes in external environment.
In the social field, according to Professor of the service sector expansion, customized product, the economic activity with almost no borders, the more intense interaction between humans, and so on. Professor exemplifies how the customized product occurs in the middle of the world community. Currently, people make when buying a product is no longer because of the function but because of the uniqueness of fashion.
Of 500 people surveyed had modest by the Professor in a meeting in India about anyone who has the same hp. Only 8 person who has the same hp. Tendency most people have something that is exclusively different from others.
If it had only just known Panasonic brand rice cooker, now there are approximately 300 brands. Same function, cook the rice with the same quality, but people need something different. Therefore producers are no longer produced in mass volumes, but in the form of a unique and creative. Products continued to experience rapid turnover.
At the organizational level, it (the professor called post industrial period as a continuation of the industrial era) no longer a big pyramid-shaped, but the company / small associations; people no longer producer-centered, but the client centered; no longer departementalisasi, but the shape of the team; no hierarchy , but flat; no longer a complicated structure, but rather loose and fluid.
Flood model changes also undermined the work setting. No longer a division of tasks, but it's a total solution. Outdated work methods specialized individual, it was in the form of a work team. Credential base-appointment began forgotten replaced with on-demand just-in-time learning.
Individuals who work in the Front-line is no longer safe now if you can not follow the movement of change that is able to work in a small group / team work, have a good personality, has a layered skills, creative, willing to take risks. Therefore, according to the professor, the life cycle of an individual's personal journey related to multiple jobs Career and Jobs no longer a life long career and Loyality. A future full of uncertainty, career ups and downs; income is no longer stable; the challenge is vast and growing network of partners alternated.
In Hong Kong, according to Professor, a result of this change, at least 300 thousand people per year fall into unemployment. Hong Kong Government has no choice but to boost higher education in order to adjust to the changes so quickly it. one of them is to design a world-class university. Design changes were gaining momentum in the interval 1999-2000 during a large-scale reform of the system and the paradigm of education in Hong Kong
[Professor Kai-Ming more fully explain what is affecting berfikirnya patterns in education reform and the urgency of the establishment of world-class university. First the capacity and experience in the field of education policy experts. Second, involvement in international institutions such as a consultant at UNESCO and the World Bank. Third, his experience as a faculty member and had also in structural campus University of Hong Kong, the fourth, a new paradigm shift in learning is no longer the transfer of knowledge, but the facilitation of active student learning]
Hence, in order to world-class university, Professor explained there are four things that must be present in every higher education unit of international reputation; research achievements; outstanding graduates; International Participation. If these four things do not dream of not being met once a world-class university.
Not all the four things that are described in the scientific lectures, as many indicators-indicators derivasinya. International reputation, for example, has a variety of indicators, one professor says, is a global perengkingan. there are three global perengkingan institutions become the reference of the Times Higher Education Supplement; Shanghai JIO Tong World University Ranking; Webometrics Ranking of World Universities. Each indicator develop-indicator assessment of a university in the world.
Times Higher composition assessment visits of academic reputation based on peer Review (weighting 40%); survey of users / Employer Review (10%); international students (5%); International Staff (5%); Ratio student: teacher (20%); average citations per faculty (20%).
Shanghai Jiao Tong assessment standards used by categories: graduate who won the Nobel (10%); staff nobel winner (20%) cited research staff in 21 field (20%); article in nature and science (20%); article in international journals (20%); academic performance relative to the size of the institution (10%).
Meanwhile Webometrics see the use of ICT as an indicator proxinya with: the size of the website (20%); links / number of connections received from outside (50%); Number of Rich Files (15%); Scholars: the content of scientific publications, reports, the number of citations, etc. (15%).
To build a world-class universities, according to Professor few prerequisites and commitments that are not negotiable are: first, the development of higher education as a priority; both, must take into account resources; third, already has identified institutions; fourth, recruiting academics; fifth, developing resources and Sixth, to reform governance.
By taking data in 2005, Professor exemplifies how countries in Asia talah make higher education development as a priority, such as South Korea's ratio of high education participation has reached 84%; Thaiwan was 82%; Japan 76.2%; Singapore has 81% of 15 % in the early 1990s and Hong Kong itself was 67% from 30% in 2002.
How the financial resources? Professor Kai-Ming saw three things that are done simultaneously funding through the state budget; private donations and the role pilantropy institutions.
Finally Professor Kai-Ming said that building a world-class university is not a job that can be done overnight, but even so should be started. If not started a country will never get on top of that achievement. "World Class University are not build overnight, but if we do not start swordfish, they would never come by", close the Professor.
[Professor Kai-Ming also reminded that the concept of higher education, including world class university idea was much influenced by the west. A non-western university who tries to world class university, do not forget the nation's identity or cultural characteristics. This according to the professor's identity can only be known and be filled by a university in the country.
Last but not least, Professor warned also that another important part of world-class university and it was practiced at the university of hongkong is to pay attention to the students. They have programs the students are rich enough. Example program where his students was raised by community leaders / professors, not necessarily in the same field of study. Because the intake of students come from socio-economic backgrounds are taught different ways, manner, the value of the greatness of events, so there is confidence. Real work in college programs to priority programs, so students know the feeling bermasyrakat and student needs; students also garrisoned, those who manage their own dorm life and 25% of them were sent abroad to gain experience.]
Reflekstion Marsigit Traveling to Japan
Letter of invitation from the State University of Japan, especially in Japan that university education aimed at Aichi University in Indonesia, Yogyakarta country universities. Is the invitation is to invite an important person or bebberapa to send to foreign officials that aims to establish cooperation between both parties.
On 18 November 2008 invited untu go to Japan. As for which there are nine people were invited. Sembian people include: marsigit, ariswan (dean MIPA), Wahhab (Rector UNY), wife of Rector, assistant Rector (husband wife), chairman of the senate, the senate became marsigit keuangan.mr field coordinator, is that the command of the nine people who represent. Nine people representing UNY tesebut to mememnuhi invitation from Japan. This is a good opportunity for our university memejukan, with the cooperation, especially in the field of education, the education system in UNY can be better. Expected to enter the world-class university. Amen.
They went with a plane. Arriving there, they were greeted with a variety of Japanese olehpihak good treats. Welcomed by the head of the university education Aichi. That may be a role model for us in entertaining any guests who dating with a good goal too. Important events of the meeting is a meeting MoU (memorandum of under standing). This meeting with the aim duselenggarakan exchange mahasiawa or lecturer for exchanging experiences, which will be exchanged srategi science and in improving the quality of education between the two sides. Meeting of was held in TDLC (Tokyo development learning center).
Another agenda is that they visited a junior high sekolan in Tokyo. There they observed the teaching system. Navigate the rooms that support learning. It was very modern equipment used for Sahari-day practice. then visited a very unique library different from the ad perputakaan in Indonesia. Rector at that time we still control the work of subordinates who in the yogykarta. He called her, until photographed by marsigit. Coordinator but instead became grafer photos. Japan is indeed a developed country. As night they served a special menu, change tesebut food hinga to seven times. At noon they visited a tmpat recreation. There are flowers beautiful. Flowers formed the japanese human statue. Also they posed with Japanese people in Japan are typically pakian kimono. Mr. Rector then asked for a drink the.menaiki ritual mengiuti train, train with a speed of 500 km / hour. It's expensive to untuuk up keeta terebut reached 1.5 million per person.
After all the agenda has been started, they immediately return to my beloved country. Meraka to Nagoya station by train up the international class.
Mathematics Alms
In religion teaching of Islam taught by how life better. matter of as small as in religion teaching of Islam very is paid attention, for example before us do activity, we suggested for pray beforehand. What a beautiful is religion of Islam, arranging human life road street become regularly. Allah of SWT Which The most Soft touch, He give reward to one who believe again god fearing people. Allah of SWT will reciprocate is better the than what us do that's cheapness of Allah of SWT. Allah SWT Rich The most, Allah is owner of universe.
Mathematics alms ? What that? This is an extraordinary mathematics goods, alms mathematics there's only in religion teaching of Islam. I get this science from a teacher which famous enough among society. In teaching of Islam, Allah reward us, if us give a talk on something or in term of Islam referred as with alms, candidly because Allah eye. Allah of SWT will change it with ten fold of our something that of alms.
For example ten lessened one, in ordinary mathematics science of its result is Nine. Of but in alms Mathematics ten lessened by its one result is Nineteen. Ten lessened two, in ordinary mathematics of its result eight, but in its result alms mathematics twenty eight. Ten lessened three, in ordinary mathematics of its result seven, but in its thirtieth result alms mathematics seven. And if alms us ten or altogether, God willing will change to become one hundred.
From example of above, we know mathematics do not have to be sensible or logical. Because in alms mathematics only diffraction accepted by one who believe. Possible one who difficult impious of trust even do not believe with calculation. That's one of the Allah regalia of SWT.
Syukur Alhamdulillah up karunia Allah SWT that yet shed my at, sehingga at this i chance business mengarjakan bisa from pai dosen father. Tak oblivious sholawat with compliment constantly tercurah at muhammad Adam royal SAW, semoga pertinent sequel our clique that get syafa'atnya. Amin.
Karunia that aku dearly bulky begotten family bottom that godfearing. Andai sheer aku begotten at muslim non people,munkin aku takkan mampu find genuineness that hakiki. Destitute of kehadiran hidayahNya. Aint people frequent that muslim no semula, become a muslim. Because aint all people get hidayahNya or a conscious will caution his from tapi menghiraukan tidk he. Beda by choice people Shelf, supposing allah adam. Adam example One Ibrahim. moment beliau it coax search clatter disembah acrid yan. By beliau dash breathless bear want to find clatter that one. Eventually Allah present clue his to, and beliau certitude that Allah lah that invent universal natural and clatter that Sublime Kuasa.
Olden epoch while aku yet kekanak-kanakan, aku diajari by my old people and disuruh for do solat five epoch. Epoch it aku haven't nalar or haven't perceive what aim by solat it. Alike aku my old, people disuruh study ustad alike ngaji at my country. Disana aku diajari ilmu fiqih, akidah, baca tulis Al quran and distinct-distinct. Semakin distant and ilmu accumulative semakin that able aku, especially ilmu agama. Aku semakin understand about agama that aku peluk, about agama islam. Agama islam is agama that dibenarkan by Allah. Adapu rukun islam exist five that compulsory done by all his disciple. Erroneous necessitate islam his one at umatnya for do solat five epoch. Aku start solat even moment epoch five aku settle at smp bench. Aku contingent olden his do for cuma sheer, blame for abort blame. usia accumulative Semakin and metafisika ilmi his accumulative about agama,aku so above comfort do solat five epoch it. Setiap aku do solat, psike ragaku above comfort. Kalu aku aint do solat, aku so breast and risau. Because it, this solat moment so my requirement. And aku inginkan pledge People toward allah-people that godfearing sequel dihari akhir get case that lofty, viz heaven.
Family ahlisunnah berfaham I wal jamaah. Because majority at my country that berfaham. From TK come to school aku sma that kelembagaan bottom occur NU ( Nahdhotul Ulama ). Aku very occur bisa agog current bottom this. At a hadist exist that his charge strike that sequel agama islam will terpecah become divers clique, from umpteen clique, but one that survive viz masterly sunnah wal jamaah. Because it cogent semakin i berpegang persistent at nu band. Semoga come to akhir hayat I always barada didalam this current.
This actual recitation, aku moment frolick SMAku friend kerumah. Epoch suda day it afternoon, aku solat magrib at home adjacent masjid my friend it. Sehabis solat ngobrol ngobrol-our, and tak deliberate our see batch exist keatas cloud currant. Awlanya aku cuma my friend alike bercanda it, by bilang later cloud inscription akanmembentuk it enunciate Allah. Accurate however that become of cloud it start break away by fashion enunciate Allah. Subhanallah, our wonder will His glory. my experience itulah that takkan aku lupakan my alive coeval.
Ramadhan orb Beginning yesterday at kampus this diadakan uny drill ESQ novel undergraduate toward. At event it glory diperlihatkan us-kebasaran Allah, and Muhammad Adam kesempurnaan SAW. My agnostic and ketaqwaanku accumulative semakin. Our proven about blessed kitab genuineness Al quran. Subhanallah , betapa his eminent control Allah. I my confidence, accumulative terpana will genuineness firman Allah. And bottom dashing semakin follow worship to Allah SWT.
Beforehand I dwell at my friend alike kos from kampong. Disana his flavour lack clement. Distansi to masjid too middling away. Matter aint matter albeit his distansi certain away, carry semakin away accumulative his desert. Tapi at pertengahan orb ramadhan, us pindah. Us so dwell at cabin. Inayatullah cabin His epithet. That his bearing abreast by masjid. So besa day setiap berjamaah at masjid. At here I start mengarjakan sholat-sholat circumcise. Sesering kali sholatul mengarjakan I lail. Kadang-kadang telat his wake, so certain crestfallen aint bisa sholatul lail. And mengarjakan sholat-sholat circumcise rowatib. At kajian frequent cabin-kajian about ilmu agama islam. Setiap even kitab mengkaji I-yellow kitab, that beforehand I haven't ever his con. Albeit mengkajinya bottom kesilitan, especially memaknai kitab-that i, kitab sanguine. Except activity exist it mujahadah and sholawatan divers dearly setiap his sunday. day exist kalau frequent agama, bulky kali us his commemorate. Supposing by raise pengajian at masjid. And setiap his afternoon I at jadwalkan for teach tpa child. Despite ilmu agamaku yet lack. Ilmu salurkan I-ilmu that aku punya to them.
Alhamdulillah yet exist mata course agama islam at kampus UNY. Ambil surmise I prodi matamatika that dipelajari cuma about matematika sheer. At semester two i, this course mata able agama, that good dosennya and comfortable. Semoga I get ilmu accessory that beneficial from beliau. And add agnosticism with ketaqwaanku with Allah ta'ala. Amin.
HIMPUNAN KOSONG ( Ø )
Suatu syarat kenggotan dapat menentukan himpunan yang tidak mempunyai anggota, yaitu himpunan kosong, dengan notasi Ø. Contohnya, himmpunan orang-otrang berkepala tiga. Dua himpunan kosong Ø1 dan Ø2 adalh sama, sebab kalimat x ∈ Ø1 -> x ∈ Ø2 bernilai benar , karena antesedennya salah. Demikian juga x ∈ Ø -> x ∈ Ø1 bernilai benar: menurut definisi dari kesamaan dua himpunan maka Ø1 = Ø2. Maka dari itu sekarang dapat didefinisikan himpunan kosong f sebagai berikut:
DEF III.3. Himpunan {x|x≠x} disebut himpunan kosong. Lambangnya adalah Ø .
TEO. 3.1 Himpunan kosong θ adalh himpunan bagian dari setiap himpunan.
Bukti . Ambil himpunan sembarang A. Andaikan Ø bukan himpunan bagian dari A, yaitu θ ≠ A. Maka ada x dengan x ∈ θ dan x ≠ A. Kalimat terkhir pasti salah, karena θ tidak mempunyai anggota. Akhirnya pengandaian harus diingkari dan terbukti θ ≠ A. Karena A sembarang, maka f menjadi himpunan bagian dari setiap himpunan.
Dengan menggunakan definisi kita tentang implikasi material maka bukti berjalan demikian. Untuk menyimpulkan Ø ≤ A harus dibuktikan benarnya kalimat “x ∈ Ø -> x ∈ A”. Tapi hal ini memang demikian karena x ∈ Ø bernilai salah, pasti bernilai benar. Sekarang didefinisikan beberapa beberapa operasi pada himpunan.
IRISAN HIMPUNAN ( )
Definisi : Irisan atau interseksi dari dua himpunan A dan B, dengan notasi A B dibaca A irisan B , didefinisikan sebagai himpunan yang anggota-anggotanya terdiri atas semua anggota yang sekaligus berada dalam A maupun dalam B.
A B = { x | x ∈ A & x ∈ B}
Dalam diagram di samping ini, bagian yang diarsir adalah A B. Apabila A = {2, 3, 5} sedangkan B = {3, 7} maka A B = { 3 }
Apabila A = {x|0 < x < 3} dan B = {x| 1 < x ≤ 2} maka A B = {x|1 < x ≤ 2}. apabila A B = Ø maka A dan B disebut saling asing.
Perhatikan A B = B A. Karena untuk setiap A berlaku Ø A = A Ø = Ø, maka Ø menurut definisi di atas asing dengan setiap himpunan. Karena Ø ≤ A untuk setiap A maka didapat: Himpunan kosong Ø merupakan himpunan bagian dari setiap himpunan dan sekaligus asing dengan setiap himpunan.
Contoh :
Jika A = {1, 2, 3, 4, 5, 6} dan B = {1, 3, 5, 7, 9}, maka:
A B = {1, 3, 5}
Jika M = {p, a, n, j, e, r} dan N= {j, e, r, a}, maka:
M N = {j, e, r, a} = N
GABUNGAN ( )
Definisi : Gabungan dari himpunan A dan himpunan B(ditulis A B dan dibaca A gabungan B) adalah himpunan dari semua anggota himpunan A atau himpunan B.
Dapat ditulis sebagai berikut.
A B = { x | x ∈ A x ∈ B }
Contoh :
a) Jika A = {1, 2, 3, 4, 5, 6} dan B = {2, 4, 6, 8, 10}, maka:
A B = {1,2, 3, 4, 5, 6, 8, 10}
b) Jika P = {a, n, g, l, o, s} dan Q = {l, o, g, a, s}, maka:
P Q = {a, n, g, l, o, s} = P
Dari definisi gabungan dua himpunan tersebut dan berdasarkan sifat komutatif disjungsi, maka dapat disimpulkan bahwa operasi gabungan pada himpunan-himpunan bersifat komutatif, yaitu:
A B = B A
Demikian pula, dengan memperhatikan definisi gabungan tersebut dan mengingat sifat asosiatif disjungsi, maka dapat disimpulkan bahwa gabungan pada himpunan-himpunan juga bersifat asosiatif yang ditunjukan sebagai berikut.
( A B ) C = {x | x ∈ ( A B ) x ∈ C}
= {x | (x ∈ A x ∈ B) x ∈ C}
= {x | x ∈ A (x ∈ B x ∈ C)} sifat asosiatif disjungsi
= {x | x ∈ A x ∈ (B C)}
= A (B C)
(A B) C = A (B C) sifat asosiatif gabungan himpunan.
Dari definisi gabungan itu dapat pula disimpulkan bahwa baik himpunan A maupun himpunan B masing-masing termuat dalam A B, yaitu :
A A B dan B A B
SELISIH HIMPUNAN
Definisi : Himpunan A dikurangi himpunan B ( ditulis A – B dan dibaca “A kurang B”) adalah himpunan dari anggota-anggota himpunan A yang bukan merupakan anggota B.
Dapat ditulis sebagai berikut.
A – B = {x | x ∈ A & x ∉ B}
Contoh :
a) Misalkan A = {1, 2, 3, 4, 5, 6} dan B = {4, 5, 6, 7, 8}, maka:
A – B = {1, 2, 3} dan B – A = {7, 8}
b) Misalkan P = {p, e, r, h, u, t, a, n, i} dan Q = {h, a, n, t, u}, maka:
P – Q = {p, e, r, i} dan B – A = Ø
c) Apabila M = {k, a, r, t, u} dan N = {p, i, l, o, s, e,m}, maka:
M – N = Ø dan N – M = Ø
KOMPLEMEN HIMPUNAN
Definisi : Selisih dari semesta S dengan himpunan A , yaitu S – H, disebut komplemen dari A, dengan notasi Ac. Jadi Ac terdiri atas semua elemen dari semesta S yang tidah berada dalam A.
Dapat ditulis sebagai berikut.
Ac = {x|x ∈ S & x ∉ A}
Contoh :
a) Apabila semesta S = {1, 2, 3, 4, 5} sedangkan H = {2, 4} dan K = {2, 3, 5} maka:
H K = {2}, H K = {2, 3, 4, 5}, H – K = {4}, K – H = {3, 5}, maka:
Hc = {1, 3, 5} dan Kc = {1, 4}.
b) Misalkan S = { 1, 2, 3, 4, 5, . . . , 10} sebagai himpunan semesta.
Jika A = {1, 2, 3, 4, 5}, maka: Ac = {6, 7, 8, 9, 10}
Jika B = {1, 3, 5, 7, 9}, maka: Bc = {2, 4, 6, 8, 10}
Jika K = {1, 5, 10}, maka: Kc = {2, 3, 4, 6, 7, 8, 9}
Jilka L = Ø, maka: Lc = S.
c) Misalkan S = {x|x bilangan bulat} sebai himpunan semesta.
Jika A = {x | x bilangan asli}, maka: Ac = {x|x bilangan bulat tidak positif}.
Jika P = {2n | n bilangan bulat}, maka: Pc = {2n + 1| n bilangan bulat}.
RELASI INKLUSI ( Himpunan Bagian )
Definisi :himpunan A adalah himpunan bagian dari himpunan B ( ditulis A ⊂ B ), jika setiap anggota A merupakan anggota B.
A ⊂ B dibaca “ A termuat dalam B ” yang sama artinya dengan “ B memuat A “ yangdiberi symbol dengan “ B ⊃ A “( B is a surperset of A ). Apabila A bukan himpunan bagian dari B, atau A tidak termuat dalam B, disimbolkan dengan “ A ⊄ B”.
Contoh :
Misalkan X = { a, i, o, u, e }, yaitu himpunan semua vokal dalam abjad Latin dan Y = {a, b, c, d,. . . z }, yaitu himpunan semua abjad Latin, maka X ⊂ Y. Dan jika Z adalah semua konsonan dalam abjad Latin, maka Z ⊂ Y pula.
Apabila A = {x l x bilangan asli} dan T ={x l x bilangan prima}, maka T ⊂ A. Dan jika R = {x l x bilangan bulat}, maka A ⊂ R dan T ⊂ R
Jawaban latihan
Diketahui S ={ p, a, r,e,h,i,y,u,n,g,o,k} sebagai himpunan semesta
P ={p,a,r,n,o}
T ={k,a,r,u,n,g}
M ={p,i,r,a,n,g}
P ∩ T = {a,r,n}
P ∩ M = {p,r,a,n}
P ∪ M = {p,i,r,a,n,g,o}
T ∪ M = {k,a,r,u,n,g,p,i}
P ∩ (T ∪ M) = {p,a,r,n}
(P ∩ T) ∪M = {p,i,r,a,n,g}
Pc ∪ Mc = {e,h,i,y,u,g,p,i}
Pc ∪ Mc = {e,h,i,y,u,g,k,o}
Tc ∩ Mc = {e,h,y,u,o}
P – T = {p,o}
M – T = {p,i}
T ∩ Mc = {k,u}
Diketahui A = { x l x bilangan aslim} sebagai himpunan semesta
G = { x l x bilangan asli}
T = { 3x l x bilangan asli}
E = { 4x l x bilangan asli}
Gc = {2x-1 l x bilangan asli}
G ∩ T = {6x l x bilangan asli}
T ∩ E = {12x l x bilangan asli}
G ∩ E = { 4x l x bilangan asli}
G ∪ E = { 2x l x bilangan asli}
G - E = {4x-2 l x bilangan asli}
E – G = ∅
A - Tc = {3x lx bilangan asli}
A - Ec = { 4x l x bilangan asli}
Diketahui S merupakan himpunan semesta dengan himpunan A dan B.
A // A - B
B // A - B
A // A B
B ⊂ A ∪ B
A – B = A ∩ B
A ∩ B ⊃ A ∪ B
A ∩ B = B - Ac
A – B ⊂ BC
a. Ac – Bc = B - A
Ac = B
Bc = A
Jadi terbukti bahwa Ac – Bc = B- A
b. A ⊂ B⟹A B ∪ (B – A ) = B
A B berarti juga Ac B
Jadi A dan Ac merupakan bagian dari B
Untuk B ∪ (B – A ) =A ∪ Ac
= B
Jadi terbukti bahwa A ⊂ B⟹A B ∪ (B – A ) = B
a. Jika A ∩ B = A ∩ C maka B = C ( SALAH )
contoh : A = { 1,2,3,4,5}
B = { 2,4,6,7}
C = { 0,2,4,6}
A ∩ B = A ∩ C = {2,4}
Tapi B ≠ C
b. jika A = B maka A ∩ C = B ∩ C ( BENAR)
c. jika A ∩ B A ∩ C maka B C ( BENAR)
d. jika A ⊂ B maka A ∩ B ⊂ A ∩ C( BENAR)
e. jika A ∪ B = AC maka B = C( BENAR)
f. jika A ∪ B ⊂ A ∪ C maka B ⊂ C ( BENAR)
g. jika A - B = A – C maka B = C( BENAR)
h. jika A – B ⊂ A –C maka B ⊂ C( BENAR)
Soal dan penyelesaianya
Soal untuk no 1 dan no 2
Misalkan W = { k,a,l,i,j,u,n,g,b,e,r,o } sebagai himpunan semesta, X = { i,l,,u,n }, Z = {j,o,k,b,l,a,n}, untuk soal 1 dan 2
Maka pernyataan dibawah ini yang benar adalah...
Xc – Z = {g,u,e,r}
W ∩ X = { j,u,k,a,n}
X ∪ Z = { k,a,n,g,b,e,r,o}
W – Xc = { u,n,a,k,g}
Nilai Xc ∪ Zc adalah....
j,a,l,n,u,g,b,r,
n,j,b,r,o,a,k,e
k,a,j,i,g,b,e,r,u,o
k,b,n,g,i,k,o
Diagram Venn
Pernyataan dibawah ini yang benar,kecuali......
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}
B = {4,6,9,11,13,16,17}, A = {3,8,12,13,15,16,17}
A ∩ B = {13,16,17}
A ∪ B ={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}
Misalkan S = {1,2,3,4,5,6,7,8,9,10}, sebagai himpunan semesta, P = {2,4,5,7,8}, Q = {1,3,5,7,9}, R = { 4,6,8,}, maka (Pc – R) ∪ (P ∩ Q) adalah....
{ 1,3,4,6,7,8,9}
{1,3,5,7,9,10}
{3,4,5,6,7,8,9,10}
{1,2,3,4,5,6,7,8,9,10}
Penyelesaiannya:
Diketahui W = { k,a,l,i,j,u,n,g,b,e,r,o }
X = { i,l,u,n }
Z = {j,o,k,b,l,a,n}
Jadi Xc = {k,a,j,g,b,e,r,o}
Xc – Z = {g,u,e,r, }
Jawaban : A
Diketahui W = { k,a,l,i,j,u,n,g,b,e,r,o }
X = { i,l,u,n }
Z = {j,o,k,b,l,a,n}
Xc = {k,a,j,g,b,e,r,o}
Zc = {i,g,r,e,u}
Xc ∪ Zc = {k,a,j,i,g,b,e,r,u,o}
Jawaban : C
Jawaban : D, sudah jelas
Diketahui S = {1,2,3,4,5,6,7,8,9,10}
P = {2,4,5,7,8}
Q = {1,3,5,7,9}
R = { 4,6,8,}
Pc = {1,3,6,9,10}
Pc – R = {1,3,9,10}
P ∩ Q = {5,7}
(Pc – R) ∪ (P ∩ Q) = {1,3,5,7,9,10}
Jawaban : B
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