Soal :
11. Tunjukan
bahwa himpunan bilangan kelipatan 2 merupakan grup terhadap a * b =
a + b
22. Tentukan
apakah
a. a * b =
a + b + 3
b. a * b =
a + b - 2ab
berupa group, monoid , atau
Semigroup.
33. Misalkan
G = { -1, 1}
Tunjukan bahwa G adalah group abel dibawah
perkalian biasa a + b = a * b
44. Diketahui
himpunan R = bilangan real tanpa -1
a
+ b = ab + a + b
tentukan sifat operasi binernya
Jawaban :
11. a * b =
a + b
-
Tertutup
jika
: a = 2 maka
: a
* b = a + b
b = 2 a * b =
2 + 2 = 4
-
Asosiatif
(a * b) *
c = a
* (b * c)
(a *
b) * c = (a
+ b) * c (a *
b) * c = a * (b + c)
= a + b
+ c = a + b + c
-
Identitas
a * e =
e *
a = a
a * e =
a
a * b =
a + b e * a e + a
= a + e
a * e =
a + e a = a
a =
a + e
e =
0
-
Invers
a -1 a -1 * a = e
a * b =
a + b Misalkan : a -1
= b
b =
-a
a * b =
a + b = 0
= a + (-a) = 0
0 = 0
-
Komutatif (abel)
a * b =
b *
a
a + b = b + a
Maka
a *
b = a + b anggota bilangan kelipatan 2 merupakan group abel
22. a. a * b =
a + b + 3
-
Asosiatif
(a *
b) * c
= a *
(b * c)
(a *
b) * c = (a
+ b + 3) * c (a *
b) * c = a * (b + c + 3)
= n * c = a
* n
= n + c
+ 3 = a + n + 3
= a + b
+ c + 6 = a + b + c + 6
-
Identitas
a * e =
e *
a = a
a * e =
a
a * b =
a + b + 3 e * a e + a
+ 3 = a + e + 3
a * e =
a + e + 3 a = a
a =
a + e
e =
-3
-
Invers
a -1 a -1 * a = e
a * b =
a + b + 3 Misalkan : a -1
= b
b =
- a - 3
a * b =
a + b +3 =
-3
= a + (-a - 3) + 3 = -3
0 
-3
-3
-
Komutatif (abel)
a
* b = b * a
a
+ b + 3 = b + a + 3
Maka a
* b = a + b + 3 merupakan monoid
abel
b. a * b =
a + b - 2ab
-
Asosiatif
(a *
b) * c
= a *
(b * c)
(a *
b) * c = (a
+ b – 2ab) * c (a *
b) * c = a * (b + c – 2 bc)
=
n *
c = a
* n
= n + c
- 2nc
= a + n – 2an
= (a + b – 2ab) + c – 2(a + b – 2ab)c = a + (b + c - 2bc) – 2a(b + c – 2bc)
= a + b + c – 2ab – 2ac – 2bc + 4abc = a + b + c – 2bc – 2ab – 2ac + 4abc
-
Identitas
a * e =
e *
a = a
a * e =
a
a * b =
a + b – 2ae e * a e + a
– 2ae = a + e – 2ae
a * e =
a + e – 2ae –
4ae + a a – 4ae
a – 4ae
a =
a + e – 2ae
e =
-2ae
-
Invers
a -1 a -1 * a = e
a * b =
a + b – 2ae Misalkan :
a -1 = b
b =
- a + 2ae
a * b =
a + b = -2ae
= a + (-a + 2ae) = -2ae
2ae -2ae
-2ae
-
Komutatif (abel)
a
* b = b * a
a
+ b – 2ab = b + a – 2ba
maka
persamaan a * b = a + b - 2ab disebut semigroup abel
33. a + b =
a * b
dengan G { -1, 1}
-
Tertutup
a + b =
a * b
= -1 * 1
= -1
-
Asosiatif
(a +
b) + c
= a +
(b + c)
(a + b) +
c = (a * b) + c
(a
+ b) +
c = a + (b * c)
=
n +
c = a + n
= (a *
b) * c = a * (b * c)
-
Identitas
a + e =
e +
a = a
a + e =
a
a + b =
a * b e + a e * a = a * e
a + e =
a * e 0 = 0
a =
a * e
e =
0
-
Invers
a -1 a -1 + a = e
a + b =
a * b Misalkan : a -1
= b
b =
1/a
a + b =
a * b = 0
= a * (1/a )
= 0
1 0
0
-
Komutatif (abel)
a + b = b + a
a * b = b * a
maka fungsi a + b =
a * b dengan G { -1, 1} bukan merupakan Group melainkan semigroup abel
44. a + b =
ab + a + b
dengan R = bilangan real
-
Tertutup
a + b =
ab + a + b a
+ b = (2*1) + 1 + 2
a
= 1 =
5
b
= 2
-
Asosiatif
(a +
b) + c
= a +
(b + c)
(a +
b) + c = (ab
+ a + b) + c
=
n +
c
= nc + n + c
= (ab
+ a + b)c + (ab + a + b) + c
= abc
+ ac + bc + ab + a + b + c
(a
+ b) +
c = a + (bc + b + c)
= a
+ n
= an
+ a + n
= a(bc + b + c)
+ a + (bc + b + c)
= abc + ac + bc + ab + a + b + c
-
Identitas
a + e =
e +
a = a
a + e =
a
a + b =
ab + a + b e + a ae + a
+ e = ae + a + e
a + e = ae
+ a + e a2e + a + e = a2e +
a + e
a =
ae + a + e
e =
ae
-
Invers
a -1 a -1 + a = e
a + b = ab
+ a + b Misalkan : a -1
= b
ab
+ b = -a
-
Komutatif (abel)
a + b = b + a
ab + a + b = ba + b + a
