Operasi Biner Matematika Informatika
Nama : Muhammad Thamrinaldi Apryan
NPM : 57414567
Kelas : 2IA03
Jur. : Teknik Informatika
Soal Operasi Biner :
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
Apakah 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 !
Soal Operasi Biner :
1.) a * b = a + b
·
Tertutup
jika : a =
2 maka : a
* b = a + b
b =
2 a * b = 2 + 2 = 4
·
Asosiatif
ó (a
* b) * c = a * (b * c)
ó (a
* b) * c = (a + b) *
c ó (a
* b) * c = a * (b +
c)
ó a + b +
c
ó a +
b + c
·
Identitas
ó a
* e = e * a = a
ó a
* e = a
ó a
* b = a +
b
ó e
* a e + a = a + e
ó a
* e = a +
e ó a = a
ó a = a + e
ó e = 0
·
Invers
ó a -1 a -1 *
a = e
ó a
* b = a +
b
Misalkan : a -1 = b
ó b = -a
ó a * b = a +
b = 0
ó a +
(-a) = 0
ó 0 = 0
·
Komutatif (abel)
ó a * b = b
* a
ó a + b = b + a
Maka, dapat disimpulkan bahwa a * b = a + b
anggota bilangan kelipatan 2 merupakan group abel.
2.a) a * b = a + b + 3
·
Asosiatif
ó (a * b) *
c = a * (b * c)
ó (a * b) *
c = (a + b + 3) * c
ó (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
ó a * e = e
* a = a
ó a * e = a
ó a * b = a + b +
3
ó e * a = e + a + 3 = a + e + 3
ó a * e = a + e +
3 ó a =
a
ó a = a + e
ó e = -3
·
Invers
a -1 a -1 *
a = e
ó a * b = a + b +
3
Misalkan : a -1 = b
ó b = - a - 3
ó a * b = a + b +3 = -3
ó a +
(-a - 3) + 3 = -3
ó 0 = -3
·
Komutatif (abel)
ó a * b = b
* a
ó a + b + 3 = b + a + 3
Maka,
dapat disimpulkan bahwa a * b = a + b + 3 merupakan monoid abel.
2.b) a
* b = a + b - 2ab
·
Asosiatif
ó (a * b) *
c = a * (b * c)
ó (a * b) *
c = (a + b – 2ab) *
c ó (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
ó a * e = e
* a = a
ó a * e = a
ó a * b = a + b –
2ae ó e * a
e + a – 2ae = a + e – 2ae
ó a * e = a + e –
2ae ó – 4ae + a ≠ a – 4ae
ó a = a + e – 2ae
ó e = -2ae
·
Invers
ó a -1 a -1 *
a = e
ó a * b = a + b –
2ae Misalkan : a -1 =
b
ó b = - a + 2ae
ó a * b = a +
b
= -2ae
ó a +
(-a + 2ae) = -2ae
ó 2ae ≠ -2ae
- Komutatif (abel)
a
* b = b * a
a + b – 2ab = b + a – 2ba
maka, dapat disimpulkan persamaan a * b = a + b - 2ab disebut
semigroup abel.
3.) a + b = a * b
dengan G { -1, 1}
·
Tertutup
ó a + b = a * b
ó -1
* 1
ó -1
·
Asosiatif
(a
+ b) + c = a + (b + c)
ó (a + b) +
c = (a * b) +
c ó (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
ó a + b = a *
b
ó e
+ a = e *
a = a * e
ó a + e = a *
e ó 0 = 0
ó a = a * e
ó 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)
ó a + b = b
+ a
ó a * b = b * a
Maka, dapat disimpulkan, fungsi a + b = a * b dengan G { -1, 1}
bukan merupakan Group melainkan semigroup abel.
4.) a + b = ab + a + b
dengan R = bilangan real
·
Tertutup
ó a + b = ab + a +
b
ó a + b = (2*1) + 1 + 2
ó a =
1
ó = 5
ó b = 2
·
Asosiatif
ó (a + b) +
c = a + (b + c)
ó (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
ó (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
ó a + e = e
+ a = a
ó a + e = a
ó a
+ b = ab + a +
b ó e + a = ae + a + e = ae + a + e
ó a
+ e = ae + a + e ó a2e + a + e = a2e
+ a + e
ó a = ae + a + e
ó e = ae
·
Invers
a -1 a -1 +
a = e
a
+ b = ab + a +
b
Misalkan : a -1 = b
ab +
b = -a
·
Komutatif (abel)
ó a + b = b
+ a
ó ab + a + b = ba + b + a
Maka, dapat disimpulkan, fungsi a + b = ab + a + b dengan P
bilangan real merupakan semigroup abel.
