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RELASI

RELASI. Definisi dan Notasi: Relasi R dari A ke B merupakan sub-himpunan dari A  B R : A  B Representasi dari relasi R : A  B, bisa dilakukan dg 4 cara: 1) himpunan pasangan terurut, 2) pemetaan, 3) matriks, 4) Notasi Digunakan matriks dengan :

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RELASI

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  1. RELASI

  2. Definisi dan Notasi: • Relasi Rdari A ke Bmerupakan sub-himpunan dari A  B • R : A  B • Representasi dari relasi R : A  B, bisa dilakukan dg 4 cara: 1) himpunan pasangan terurut, 2) pemetaan, 3) matriks, 4) Notasi • Digunakan matriks dengan : • baris merepresentasikan elemen-elemen A • kolom merepresentasikan elemen-elemen B • entri (ai, bj) = 1 jika (ai, bj)  R, i,j menunjukkan indeks • entri (ai, bj) = 0 jika (ai, bj)  R

  3. Contoh: • A = { a, p, x }; B = { b, q, y, z }, AxB={(a,b), (a,y), (a,q), (a,z), (p,b), …} , R1={(a,z), (p,y), (p,b)} • R = { (a, b), (p, q), (x, y), (x, z) } • (a, b) R atau a R b • (a, q) R atau a R q Relasi R dlm bentuk matriks Relasi R dlm bentuk pemetaan b q y z b q y z a p x a p x 10 0 0 0 10 0 0 01 1

  4. Invers dari relasi R (R–1), R–1 : B  A R–1 = { (b,a) | (a, b)  R} = { (b, a), (q, p), (y, x), (z, x) } Komplemen dari relasi R, R : A  B R = { (a, b | (a, b)  R , tapi (a,b)  AXB} = { (a, q), (a, y), (a, z), (p, b), (p, y), (p, z), (x, b), (x, q) } Tentukan himp relasi yang unsur absisnya huruf vokal dan ordinatnya huruf konsonan dari A dan B di atas!

  5. R={(a,b), (a,q), (a,y), (a,z)}

  6. Relasi pada sebuah himpunan (relation on a set) • R : A  Aadalah sub-himpunan dari A  A • Contoh (Example 5): R : Bil Bulat  Bil Bulat • R1 = { (a, b) | a  b} • R2 = { (a, b) | a  b} • R3 = { (a, b) | a = b or a = – b } • R4 = { (a, b) | a = b} • R5 = { (a, b) | a = b + 1 } • R6 = { (a, b) | a + b  3}

  7. Relasi pada sebuah himpunan (relation on a set) • R : A  Aadalah sub-himpunan dari A  A • Representasi dari R : A  A • Menggunakan Matriks Relasi (banyaknya baris = banyaknya kolom) • Menggunakan Directed Graph (disingkat Digraph=Graph berarah) • Contoh : A = { 1, 2, 3 }; R = { (1, 1), (1, 2), (2, 3), (3, 1) } • 2 • 1 • 3 1 10 0 01 10 0

  8. Sifat-sifat relasi R : A  A • Refleksif : a [ (a, a)  R ] • Irefleksif : a [ (a, a)  R ] • Sifat-sifat relasi R : A  B • Simetrik : a,b [ (a, b)  R  (b, a)  R ] • Antisimetrik : a,b [ ((a, b)  R  (b, a)  R)  (a = b) ] • atau a ,b [ (a  b)  ((a, b)  R  (b, a)  R) ] • 3. Transitif: a,b,c [((a, b)  R  (b, c)  R)  (a, c)  R ] • 4. Asimetrik : a, b [ (a, b)  R  (b, a)  R ]

  9. Contoh (Example 5): Cek sifat-sifat relasi R : A  A , di mana A = { 1, 2, 3, 4 } R1 = { (a, b) | a  b} R4 = { (a, b) | a = b} R2 = { (a, b) | a  b} R5 = { (a, b) | a = b + 1 } R3 = { (a, b) | a = b or a = – b } R6 = { (a, b) | a + b  3} R1 = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4)} Refleksif : ya, karena(1,1), (2,2), (3,3), (4,4)  R1 Irefleksif : tidak, karena (1,1)  R1 Simetrik : tidak, karena (1, 3)  R1 (3, 1)  R1 Asimetrik : ya, karena (4a, 4b)  R1 (4b, 4a)  R1 Antisimetrik : ya, karena [(4a, 4b)  R1 (4b, 4a)  R1]  ( 4a = 4b) memenuhi untuk (1,1), (2,2), (3,3) juga

  10. Contoh (Example 5): Cek sifat-sifat relasi R : A  A , di mana A = { 1, 2, 3 } R2 = { (a, b) | a  b} R5 = { (a, b) | a = b + 1 } R6 = { (a, b) | a + b  3} Periksa sifat relasi utk relasi tsb Kerjakan per kelompok maks 3 org Sekarang!

  11. R1 : { 1, 2, 3, 4 }  { 1, 2, 3, 4 }, A di mana R1 = { (a, b) | a  b} R1 = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4)} Transitif : (a, b)  R1 dan (b, c)  R1  (a, c)  R1 (1,1)  R1 dan (1,1)  R1  (1,1)  R1 ; (1,1)  R1 dan (1,2)  R1  (1,2)  R1 (1,1)  R1 dan (1,3)  R1  (1,3)  R1 ; (1,1)  R1 dan (1,4)  R1  (1,4)  R1 (1,2)  R1 dan (2,2)  R1  (1,2)  R1 ; (1,2)  R1 dan (2,3)  R1  (1,3)  R1 (1,2)  R1 dan (2,4)  R1  (1,4)  R1 ; (1,3)  R1 dan (3,3)  R1  (1,3)  R1 (1,3)  R1 dan (3,4)  R1  (1,4)  R1 ; (1,4)  R1 dan (4,4)  R1  (1,4)  R1 (2,2)  R1 dan (2,2)  R1  (2,2)  R1 ; (2,2)  R1 dan (2,3)  R1  (2,3)  R1 (2,2)  R1 dan (2,4)  R1  (2,4)  R1; (2,3)  R1 dan (3,3)  R1  (2,3)  R1 (2,3)  R1 dan (3,4)  R1  (2,4)  R1; (2,4)  R1 dan (4,4)  R1  (2,4)  R1 (3,3)  R1 dan (3,3)  R1  (3,3)  R1 ; (3,3)  R1 dan (3,4)  R1  (3,4)  R1 (4,4)  R1 dan (4,4)  R1  (4,4)  R1

  12. Refleksif : a [ (a, a)  R ] • R : integer  integer • R1 = { (a, b) | a  b} ya • R2 = { (a, b) | a  b} tidak • R3 = { (a, b) | a = b or a = – b } ya • R4 = { (a, b) | a = b} ya • R5 = { (a, b) | a = b + 1 } tidak • R6 = { (a, b) | a + b  3 } ya

  13. 2.Simetrik: a b [ (a, b)  R  (b, a)  R ] • R : integer  integer • R1 = { (a, b) | a  b} tidak • R2 = { (a, b) | a  b} tidak • R3 = { (a, b) | a = b or a = – b } ya • R4 = { (a, b) | a = b} ya • R5 = { (a, b) | a = b + 1 } tidak • R6 = { (a, b) | a + b  3 } ya

  14. 3. Antisimetrik : a b [ ((a, b)  R  (b, a)  R)  (a = b) ] • atau a b [ (a  b)  ((a, b)  R  (b, a)  R) ] • R : integer  integer • R1 = { (a, b) | a  b} • R2 = { (a, b) | a  b} • R3 = { (a, b) | a = b or a = – b } • R4 = { (a, b) | a = b} • R5 = { (a, b) | a = b + 1 } • R6 = { (a, b) | a + b  3 }

  15. 4. Transitif : • abc [((a, b)  R  (b, c)  R)  (a, c)  R ] • R : integer  integer • R1 = { (a, b) | a  b} • R2 = { (a, b) | a  b} • R3 = { (a, b) | a = b or a = – b } • R4 = { (a, b) | a = b} • R5 = { (a, b) | a = b + 1 } • R6 = { (a, b) | a + b  3 }

  16. 5. Irefleksif : a [ (a, a)  R ] • R : integer  integer • R1 = { (a, b) | a  b} • R2 = { (a, b) | a  b} • R3 = { (a, b) | a = b or a = – b } • R4 = { (a, b) | a = b} • R5 = { (a, b) | a = b + 1 } • R6 = { (a, b) | a + b  3 }

  17. 6. Asimetrik : a b [ (a, b)  R  (b, a)  R ] • R : integer  integer • R1 = { (a, b) | a  b} • R2 = { (a, b) | a  b} • R3 = { (a, b) | a = b or a = – b } • R4 = { (a, b) | a = b} • R5 = { (a, b) | a = b + 1 } • R6 = { (a, b) | a + b  3 }

  18. SOAL: • Periksa ke-6 sifat relasi untuk • Relasi invers dari R1 s/d. R6 • Relasi komplementer dari R1 s/d. R6 • Catatan: R : A  B • Relasi invers dari R, notasi R-1: B  A • { (b, a) | (a, b)  R } • Relasi komplemen dari R, notasi R: A  B • { (a, b) | (a, b) R }

  19. Kombinasi dua relasi: • R1 : A  B • R2 : A  B • R1  R2 • R1  R2 • R1  R2 • R1 – R2 • R2 – R1 • Catatan: baca Examples 3, 4 (halaman 491, 492)

  20. Komposisi dua relasi: A B C a b c R : A  B S : B  C dan disebut relasi komposit/komposisi R S S  R Komposisi ditulis sebagai S  R

  21. Contoh: R : A  B di mana A = { 1, 2, 3 } dan B = { 1, 2, 3, 4 } S : B  C di mana C = { 0, 1, 2 } R = { (1,1), (1,4), (2,3), (3,1), (3,4) } S = { (1,0), (2,0), (3,1), (3,2), (4,1) } S  R = ………………. Soal: Gambarkan relasi komposit tersebut.

  22. Representasi relasi komposit: R : A  B di mana A = { 1, 2, 3 } dan B = { 1, 2, 3, 4 } S : B  C di mana C = { 0, 1, 2 } R = { (1,1), (1,4), (2,3), (3,1), (3,4) } S = { (1,0), (2,0), (3,1), (3,2), (4,1) } MR = MS = MS°R = MR MS (perkalian Boolean MR dan MS)

  23. MR = MS = MS°R = MR MS (perkalian Boolean MR dan MS) =1 1 0 0 1 1 1 1 0

  24. Komposisi lebih dari dua relasi R: A  A • R1 = R • R2 = R  R • R3 = R2 R • ………. • Rn+1 = Rn  R

  25. Contoh: R : {1, 2, 3, 4}  {1, 2, 3, 4} R = { (1,1), (2,1), (3,2), (4,3) } R2 = { (1,1), (2,1), (3,1), (4,2) } R3 = { (1,1), (2,1), (3,1), (4,1) } R4 = { (1,1), (2,1), (3,1), (4,1) } R5 = { (1,1), (2,1), (3,1), (4,1) } dst Soal: Verifikasi dengan gambar

  26. R : {1, 2, 3, 4}  {1, 2, 3, 4} R = { (1,1), (2,1), (3,2), (4,3) } R2 = { (1,1), (2,1), (3,1), (4,2) } R3 = { (1,1), (2,1), (3,1), (4,1) } MR = MR = MRMR 2

  27. RELASI n-ary Sub-bab 7.2

  28. Relasi R: • Binary : (a1, a2) disebut ordered-pair • Contoh : (Nama_mahasiswa, nilai_UTS) • Ternary : (a1, a2, a3) disebut ordered-triple • Contoh : (NRP_mhs, Nama_mhs, nilai_UTS) • Contoh lain: • R adalah relasi (penerbangan, no-penerbangan, asal, tujuan, waktu-berangkat) • Disebut quintuple (karena terdiri dari 5 komponen) • n-ary : (a1, a2, a3, … , an) disebut n-tuple

  29. Relasi R: • Binary : (a1, a2) disebut ordered-pair • Contoh : (Nama_mahasiswa, nilai_UTS) • Ternary : (a1, a2, a3) disebut ordered-triple • Contoh : (NRP_mhs, Nama_mhs, nilai_UTS) • Contoh lain: • R adalah relasi (penerbangan, no-penerbangan, asal, tujuan, waktu-berangkat) • Disebut quintuple (karena terdiri dari 5 komponen) • n-ary : (a1, a2, a3, … , an) disebut n-tuple

  30. Definisi: • Relasi n-ary adalah sub-himpunan dari A1  A2 A3 …  An • Himpunan-himpunan A1, A2, A3, …, An disebut domaindari relasi • n disebut derajatrelasi • Aplikasi: Basis Data Relasional

  31. Terminologi: Tabel : alternatif representasi basis data relasional Primary-key : a domain of an n-ary relation such that an n-tuple is uniquely determined by its value for this domain Composite-key : the Cartesian product of domains of an n-ary relation such that an n-tuple is uniquely determined by its values for these domains Projection : a function that produces relations of smaller degree from an n-ary relation by deleting fields Join : a function that combines n-ary relations that agree on certain fields SQL : Structured Query Language

  32. Primary-key :a domain of an n-ary relation such that an n-tuple is uniquely determined by its value for this domain • Contoh: lihat Tabel 1 • 4-tuple : (nama, nomor-identitas, jurusan, IPK) • (Ackermann, 231455, CS, 3.88) • (Adams, 8888323, Physics, 3.45) • (Chou, 102147, CS, 3.49) • (Goodfriend, 453876, Math, 3.45) • (Rao, 678543, Math, 3.90) • (Stevens, 786576, Psychology, 2.99) • Alternatif primary-key: nama, nomor-identitas

  33. Composite-key :the Cartesian product of domains of an n-ary relation such that an n-tuple is uniquely determined by its values for these domains • Contoh:4-tuple : (nama, nomor-identitas, jurusan, IPK) • (Ackermann, 231455, CS, 3.88) CS, 3.45 • (Adams, 8888323, Physics, 3.45) CS, 3.88 • (Chou, 102147, CS, 3.49) Math, 3.45 • (Goodfriend, 453876, Math, 3.45) Math, 3.90 • (Rao, 678543, Math, 3.90) Physics, 3.45 • (Stevens, 786576, Psychology, 2.99) Psychology, 2.99 • Alternatif composite-key: jurusan x IPK

  34. Projection : a function that produces relations of smaller degree from an n-ary relation by deleting fields Pi1,i2,i3, … ,imdeletes n–m of the components of the n-tuple, leaving the i1th, i2th, i3th, …, imth components Lihat Example 7

  35. Join : Jp Jp is a function that combines all m-tuples of the first relation with all n-tuples of the second relation, where the last p components of the m-tuples agree with the first p components of the n-tuples. Lihat Example 9 halaman 486

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