1. The document discusses different types of algebraic structures, beginning with groupoids where a non-empty set G has a binary operation * that satisfies closure. 2. Semigroups are then introduced where the binary operation is associative but not necessarily having an identity element or inverses. 3. Groups are defined as sets with a binary operation that is closed, associative, has an identity element, and every element has an inverse. Examples of groups include the integers under addition. 4. Abelian groups, also known as commutative groups, are groups where the binary operation is commutative. Matrix multiplication of 3x3 matrices forms an Abelian group.

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 I
Dosen Pengasuh : 1. Dr. Darmawijoyo
2. Dr. Nila Kesumawati, M.Si.

2. O GRUPOID
O SEMIGRUP
O GRUP
O GRUP ABEL

3. GRUPOID
Definisi 1.2.1
Suatu himpunan tidak kosong, G
dengan operasi biner (*) didalamnya,
disebut grupoid dan dinyatakan
dengan (G,*)

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.

5. SEMIGRUP

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

11. Contoh 3:
Penyelesaian:
x -1 1
-1 1 -1
1 -1 1

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

16. GRUP ABEL

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 .

18. Terima Kasih

19. Elements Page