4. Contoh 1:
* x y z
x x y y
y y x y
z z y x
Ta 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 n
gr 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 r
da l a m G.
6. Contoh 2:
Mi s a l k a n h i mp u n a n
b i l a n g a n a s l i
N, d i d e f i n i s i k a n o p e r a s i
b i n e r : a *b = a + b + a b
T 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. 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. Penyelesaian:
J a d i , (N ,*) m e r u p a k a n s u a t u s e m
9. GRUP
Definisi 1.2.3
Suatu himpunan tidak kosong G
merupakan suatu grup, jika dalam
G terdapat operasi misalkan * dan
unsur-unsur dalam G memenuhi
syarat:
10. Grup
1. T e r t u t u p
2. A s s o s i a t i f
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. 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. 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. 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
17. Contoh 4:
Penyelesaian:
-1 x 1 = -1 d a n 1 x (-1) = -1
s e h i n g g a -1 x 1 = 1 x (-1) = -1
J a d i , (G ,x ) m e r u p a k a n g r u p
k o mu t a t i f a t a u g r u p
a b e l .
