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Himpunan dan Relasi Fuzzy

Himpunan dan Relasi Fuzzy. Bahan Kuliah IF4058 Topik Khusus IF. Oleh : Rinaldi Munir. Teknik Informatika – STEI ITB. Operasi pada Himpunan Tegas. Gabungan ( union ) A  B = { x | x  A atau x  B}  A  B =  A (x)   B (x) = max(  A (x),  B (x))

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Himpunan dan Relasi Fuzzy

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  1. HimpunandanRelasi Fuzzy BahanKuliah IF4058 TopikKhusus IF Oleh: RinaldiMunir TeknikInformatika – STEI ITB

  2. OperasipadaHimpunanTegas • Gabungan (union) A  B = { x | x  A atau x  B} AB = A(x) B(x) = max(A(x), B(x)) • Irisan (intersection) A  B = { x | x  A dan x  B } AB(x) = A(x) B(x) = min(A(x), B(x)) • Komplemen A’ = { x | x  A, x  X } A’(x) = 1 - A(x) • Perkaliankartesian (cartesian product) A  B = { (a,b) | a  A dan b  B } • Selisih (difference) A – B = { x | x  A dan x  B } = A  B’

  3. OperasipadaHimpunan Fuzzy • Misalkanhimpunanfuzzy A danhimpunan fuzzy B masing-masingmemilikifungsikeanggotaan yang grafiknyaadalahsebagaiberikut:

  4. Gabungan • ABAB = A(x) B(x) = max(A(x), B(x)) • ABdiartikansebagai “xdekatAatauxdekatB”.

  5. Irisan • ABAB = A(x) B(x) = min(A(x), B(x)) • ABdiartikansebagai “xdekatAdanxdekatB”.

  6. Komplemen  1 – A(x) • diartikansebagai “xtidakdekatA”.

  7. Sifat-sifatHimpunanTegas • Komutatif • A  B = B  A • A  B = B  A • Asosiatif • A  (B  C) = (A  B)  C • A  (B  C) = (A  B)  C • Distributif • A  (B  C) = (A  B)  (A  C) • A  (B  C) = (A  B)  (A  C) • Idempoten • A  A = A • A  A = A

  8. Identitas • A  = A • A  X = A • Involusi • (A’)’ = A • De Morgan • (A  B)’ = A’  B’ • (A  B)’ = A’  B’ • Null • A  =  • A  X = X

  9. Sifat-sifatHimpunan Fuzzy • Sifat-sifathimpunan fuzzy samadengansifathimpunantegas. • Tetapi, adabeberapapengecualiansebagaiberikut: (a) Padahimpunantegas: A  A’ = X A  A’ = 

  10. (b) Padahimpunan fuzzy A  A’  X A  A’ 

  11. Relasi Fuzzy • Relasiadalahasosiasiantaraduaataulebihobyekdariduabuahhimpunan. • Contoh: ‘s lebihkecildari t’ adalahcontohrelasibiner. • Relasipadahimpunantegas Contoh: R(s,t) adalahrelasipadahimpunan S dan T, s  S, t  T, yang berarti “s lebihkecildaripada t” S = {1, 2, 5}; T = {2, 3}; R= {(1, 2), (2, 3) } t 2 3 s 1 1 0 R(s,t) = 2 0 1 5 0 0

  12. Relasipadahimpunanfuzzy Relasifuzzymemetakanelemendarisemesta X kesemesta lain Y denganmenggunakanperkaliankartesiandariduabuahsemesta. Misal: A himpunanfuzzypadasemesta X B himpunanfuzzypadasemesta Y Relasifuzzy R: R = {(x,y), R(x,y) | (x, y)  A  B } R(x,y) = A  B (x,y) = min(A(x), B (y) )

  13. Contoh: Misal x, y bilanganriildanrelasi R adalahrelasi “x dianggaplebihbesardaripada y”   0 , jika x  y R(x,y) = (x – y)/(10y) , jika y < x < 11y 1 , jika x  11y Contoh: Misal x, y bilanganbulatdanrelasi R adalah “x dianggaplebihbesardaripada y” X = {x1, x2, x3} Y = {y1, y2, y3, y4} y1 y2 y3 y4 x1 0.8 1.0 0.1 0.7 x2 0.0 0.8 0.0 0.0 x3 0.9 1.0 0.7 0.8

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