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# Grup

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### Grup

1. 1. STRUKTUR ALJABAR GRUP Oleh: F E L I R A MU R Y T R I MU H T I H A R Y A N IDosen Pengasuh : 1. Dr. Darmawijoyo 2. Dr. Nila Kesumawati, M.Si.
2. 2. O GRUPOIDO SEMIGRUPO GRUPO GRUP ABEL
3. 3. GRUPOIDDefinisi 1.2.1Suatu himpunan tidak kosong, Gdengan operasi biner (*) didalamnya,disebut grupoid dan dinyatakandengan (G,*)
4. 4. Contoh 1: * x y z x x y y y y x y z z y xTa be l i ni di ba c a x * x = x , x *y = y , z * z = x da n s e t e r us ny a(G ,*) i n i me r u p a k a ngr up oi d, k a r e na ope r a s i *me r u pa k a n ope r a s i bi ne rda l a m G.
5. 5. SEMIGRUP
6. 6. Contoh 2:Mi s a l k a n h i mp u n a nb i l a n g a n a s l iN, d i d e f i n i s i k a n o p e r a s ib i n e r : a *b = a + b + a bT u n j u k k a n b a h w a (N ,*)Penyelesaian:a 1. T e r th u st eu m i g r u p ! d a l a p J a d i , N t e r t u t u p t e r h a d a p o p
7. 7. Penyelesaian:2. A s s o s i a t i f(a * b ) * c = (a + b + a b ) * c = (a +b +a b ) + c + (a + b + a b ) c = a + b + a b + c + a c + b c + a b a * (b * c ) = a * (b + c + b c ) = a + (b +c +b c ) + a (b + c + b c ) = a + b + c + b c + a b + a c + a b
8. 8. Penyelesaian: J a d i , (N ,*) m e r u p a k a n s u a t u s e m
9. 9. GRUPDefinisi 1.2.3Suatu himpunan tidak kosong Gmerupakan suatu grup, jika dalamG terdapat operasi misalkan * danunsur-unsur dalam G memenuhisyarat:
10. 10. Grup1. T e r t u t u p2. A s s o s i a t i f
11. 11. Contoh 3:Penyelesaian: x -1 1 -1 1 -1 1 -1 1
12. 12. Penyelesaian:a . Te r t u t u p G t e r t u t u p t e r h a d a p o p e r a s i p e r k a l i a n b i a s a x k a r e n a
13. 13. Penyelesaian:b . As s o s i a t i f (a x b) x c = (-1 x -1) x 1 = 1 x 1 = 1 a x (b x c) = -1 x (-1 x 1) = -1 x -1 = 1 s e h i n g g a (a x b ) x c = a x (b x c ) = 1 m a k a G a s s o s i a t i f
14. 14. Penyelesaian:c . A d a n y a e l e m e n i d e n t i t a s (e = p e r k a l i a n A mb i l s e mb a r a n g n i l a i d a r i G -1 x e = e x (-1) = -1 1xe=ex1=1 Ma k a G me mp u n y a i i d e n t i t a s
15. 15. Penyelesaian:d . Ad a n y a i n v e r s - A mb i l s e mb a r a n g n i l a i d a r i G, - A mb i l s e mb a r a n g n i l a i d a r i G, Ma k a a d a i n v e r s u n t u k s e t i a p
16. 16. GRUP ABEL
17. 17. Contoh 4:Penyelesaian:-1 x 1 = -1 d a n 1 x (-1) = -1s e h i n g g a -1 x 1 = 1 x (-1) = -1J a d i , (G ,x ) m e r u p a k a n g r u pk o mu t a t i f a t a u g r u pa b e l .
18. 18. Terima Kasih
19. 19. Elements Page