Tugas 5 Bravo 2103015233 Aljabar Boolean
Nama: Bravo
Marvel
Kelas: 2B
NIM:
2103015233
Rangkuman Aljabar Boolean
Aljabar
boolean adalah
aljabar yang berhubungan dengan variabel biner dan operasi logika, dimana
aljabar boolean adalah sistem matematika yang terbentuk dari 3 operator logika
berupa "negasi", Logika "AND" dan "OR"
Selain
simbol logika "0" dan "1" yang digunakan untuk
merepresentasikan input atau output digital, kita juga dapat menggunakannya
sebagai konstanta pada rangkaian terbuka atau rangkaian tertutup secara
permanen.
Variabel
yang digunakan dalam Aljabar Boolean hanya memiliki dua kemungkinan yaitu
logika "0" dan logika "1" tetapi ekspresi jumlah variabel
yang dihasilkan tak terbatas yang semuanya dilabeli secara individual untuk
mewakili input ke ekspresi.
Sebagai
contoh, sebuah variabel A , B, C dll, dihasilkan sebuah ekspresi logis yaitu A
+ B = C, tetapi setiap variabel HANYA dapat berupa 0 atau 1.
Laws
& Rules of Boolean Algebra
Commutative law of addition
Commutative law
of addition,
A+B = B+A
the order of
ORing does not matter
Commutative law of Multiplication
Commutative law
of Multiplication
AB = BA
the order of
ANDing does not matter.
Associative law of addition
Associative law
of addition
A + (B
+ C) = (A + B) + C
The grouping
of ORed variables does not matter
Associative law of multiplication
Associative law
of multiplication
A(BC) =
(AB)C
The grouping
of ANDed variables does not matter
Distributive Law
A(B +
C) = AB + AC
(A+B)(C+D)
= AC + AD + BC + BD
Boolean Rules
1) A + 0 =
A
l In math if you add 0 you have changed
nothing
l In Boolean Algebra ORing with 0 changes
nothing
2) A + 1 =
1
l ORing with 1 must give a 1 since if any
input is 1 an OR gate will give a 1
3) A • 0 =
0
l In math if 0 is multiplied with anything
you get 0. If you AND anything with 0
you get 0
4) A • 1 =
A
l ANDing anything with 1 will yield the
anything
5) A + A =
A
l ORing with itself will give the same result
6) A + A =
1
l Either A or A must be 1 so A + A =1
7) A • A =
A
l ANDing with itself will give the same
result
8) A • A =
0
l In digital Logic 1 =0 and 0 =1, so
AA=0 since one of the inputs must be 0.
9) A = A
l If you not something twice you are
back to the beginning
10) A + AB
= A
Proof:
A + AB = A(1
+B) DISTRIBUTIVE LAW
= A·1 RULE 2: (1+B)=1
= A RULE 4: A·1 = A
11) A + AB = A + B
l If A is 1 the output is 1 , If A is 0
the output is B
Proof:
A + AB = (A
+ AB) + AB RULE 10
= (AA +AB) + AB
RULE 7
= AA + AB + AA
+AB RULE 8
= (A + A)(A +
B) FACTORING
= 1·(A + B) RULE 6
= A+B RULE 4
12) (A + B)(A
+ C) = A + BC
PROOF
(A + B)(A +C)
= AA + AC +AB +BC DISTRIBUTIVE
LAW
= A + AC + AB + BC RULE 7
= A(1 + C) +AB
+ BC FACTORING
= A.1 + AB +
BC
RULE 2
= A(1 + B) +
BC
FACTORING
= A.1 + BC
RULE 2
= A + BC RULE 4
Source:
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