Operasi
Biner Matematika Informatika
Soal :
1. Tunjukan bahwa himpunan bilangan kelipatan
2 merupakan grup terhadap a * b = a + b
2. Tentukan apakah :
a. a
* b = a + b + 3
b. a
* b = a + b - 2ab
berupa group,
monoid , atau Semigroup.
3. Misalkan G = { -1, 1}
Tunjukan bahwa G adalah group abel dibawah
perkalian biasa a + b = a * b
4. Diketahui himpunan R = bilangan real tanpa
-1
a
+ b = ab + a + b
Tentukan sifat operasi binernya
Jawaban :
1. a
* b = a + b
- Tertutup
jika : a = 2 maka : a
* b = a + b
b = 4
a * b =
2+ 4 = 6
- 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 = 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 karena bersifat
asosiatif,ivers,tertutup dan identitas
2. a. a
* b = a + b + 3
-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 =
e
a * b =
a + b + 3 Misalkan : a -1
= b
b = - a - 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
- 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 = a + e – 2ae
e = -2ae
- Invers
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
- 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 =
e
a + b =
a * b Misalkan : a -1
= b
b = 1/a
a + b =
a * b = 0
= a * (1/a ) = 0
1
0
- Komutatif (abel)
a
+ b = b + a
a * b
= b * a
maka fungsi
a +
b = a * b dengan G { -1, 1} adalah semigroup abel
4. 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 = 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
Maka fungsi
a +
b = ab + a + b dengan P bilangan real merupakan semigroup abel
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