tugas 7 (up/down counter)
RANGKAIAN UP/DOWN COUNTER
Rangkaian Up/Down Counter merupakan gabungan dari Up Counter dan Down Counter. Rangkaian ini dapat menghitung bergantian antara Up dan Down karena adanya input eksternal sebagai control yang menentukan saat menghitung Up atau Down.
Pada gambar 4.4 ditunjukkan rangkaian Up/Down Counter Sinkron 3 bit. Jika input CNTRL bernilai ‘1’ maka Counter akan menghitung naik (UP), sedangkan jika input CNTRL bernilai ‘0’, Counter akan menghitung turun (DOWN).
A | B | C | CARRY | SUM |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 0 | 1 |
0 | 1 | 1 | 1 | 0 |
1 | 0 | 0 | 0 | 1 |
1 | 0 | 1 | 1 | 0 |
1 | 1 | 0 | 1 | 0 |
1 | 1 | 1 | 1 | 1 |
Dalam hal ini SUM sama dengan A XOR B XOR C, dan keluaran CARRY sama dengan AB OR AC OR BC.
Keluaran SUM akan menjadi 1 apabila masukan angka 1 berjumlah ganjil.
Keluaran CARRY akan menjadi 1 apabila dua atau lebih dari dua masukkannya bernilai 1.
Gambar Rangkaian
I. Hukum Aljabar Boolean
1. HUKUM KOMUTATIF
a. A + B = B + A
A | B | A + B | B + A |
0 0 1 1 | 0 1 0 1 | 0 1 1 1 | 0 1 1 1 |
b. A . B = B . A
A | B | A . B | B . A |
0 0 1 1 | 0 1 0 1 | 0 0 0 1 | 0 0 0 1 |
2. HUKUM ASOSIATIF
a. (A + B) + C = A +(B + C)
A | B | C | A + B | B + C | (A + B) + C | A + (B + C) |
0 0 0 0 1 1 1 1 | 0 0 1 1 0 0 1 1 | 0 1 0 1 0 1 0 1 | 0 0 1 1 1 1 1 1 | 0 1 1 1 0 1 1 1 | 0 1 1 1 1 1 1 1 | 0 1 1 1 1 1 1 1 |
b. (A . B) . C = A . (B . C)
A | B | C | A . B | B . C | (A . B) . C | A . (B . C) |
0 0 0 0 1 1 1 1 | 0 0 1 1 0 0 1 1 | 0 1 0 1 0 1 0 1 | 0 0 0 0 0 0 1 1 | 0 0 0 1 0 0 0 1 | 0 0 0 0 0 0 0 1 | 0 0 0 0 0 0 0 1 |
3. HUKUM DISTRIBUTIF
a. A . (B + C) = A . B + A . C
A | B | C | B + C | A . B | A . C | A.(B+C) | A.B + A.C |
0 0 0 0 1 1 1 1 | 0 0 1 1 0 0 1 1 | 0 1 0 1 0 1 0 1 | 0 1 1 1 0 1 1 1 | 0 0 0 0 0 0 1 1 | 0 0 0 0 0 1 0 1 | 0 0 0 0 0 1 1 1 | 0 0 0 0 0 1 1 1 |
b. A + (B . C) = (A + B)(A + C)
A | B | C | B . C | A + B | A + C | A+(B.C) | (A+B)(A+C) |
0 0 0 0 1 1 1 1 | 0 0 1 1 0 0 1 1 | 0 1 0 1 0 1 0 1 | 0 0 0 1 0 0 0 1 | 0 0 1 1 1 1 1 1 | 0 1 0 1 1 1 1 1 | 0 0 0 1 1 1 1 1 | 0 0 0 1 1 1 1 1 |
4. HUKUM IDENTITY
a. A + A = A
A + A | A |
0 0 1 1 | 0 0 1 1 |
b. A . A = A
A . A | A |
0 0 1 1 | 0 0 1 1 |
5. a. A . B + A . B’ = A
A | B | B’ | A . B | A . B’ | A.B + A.B’ |
0 0 1 1 | 0 1 0 1 | 1 0 1 0 | 0 0 0 1 | 0 0 1 0 | 0 0 1 1 |
b.(A +B)(A + B’) = A
A | B | B’ | A + B | A + B’ | (A+B)(A+B’) |
0 0 1 1 | 0 1 0 1 | 1 0 1 0 | 0 1 1 1 | 1 0 1 1 | 0 0 1 1 |
6. HUKUM REDUDANSI
a. A + A . B = A
A | B | A . B | A + A .B |
0 0 1 1 | 0 1 0 1 | 0 0 0 1 | 0 0 1 1 |
b. A (A + B) = A
A | B | A + B | A (A+B) |
0 0 1 1 | 0 1 0 1 | 0 1 1 1 | 0 0 1 1 |
7. a. 0 + A = A
A | 0 + A |
0 1 | 0 1 |
b.0 .A = 0
A | 0 . A | 0 |
0 1 | 0 0 | 0 0 |
8. a. 1 + A = 1
A | 1 + A | 1 |
0 1 | 0 0 | 1 1 |
b. 1 . A = A
A | 1 . A |
0 1 | 0 1 |
9. a. A’ + A =1
A | A’ | A’ + A | 1 |
0 1 | 1 0 | 1 1 | 1 1 |
b.A’ .A =0
A | A’ | A’ . A | 0 |
0 1 | 1 0 | 0 0 | 0 0 |
10. a. A + A’ . B = A + B
A | B | A’ | A’ . B | A + A’.B | A + B |
0 0 1 1 | 0 1 0 1 | 1 1 0 0 | 1 0 1 0 | 0 1 1 1 | 0 1 1 1 |
b.A(A’ +B) = A . B
A | B | A’ | A’ + B | A (A’+B) | A . B |
0 0 1 1 | 0 1 0 1 | 1 1 0 0 | 1 1 0 1 | 0 0 0 1 | 0 0 0 1 |
11. THEOREMA DE MORGAN
a. (A + B)’ =A’ . B’
A | B | A’ | B’ | A + B | (A +B)’ | A’ . B’ |
0 0 1 1 | 0 1 0 1 | 1 1 0 0 | 1 0 1 0 | 0 1 1 1 | 1 0 0 0 | 1 0 0 0 |
b. (A .B)’ = A’ + B’
A | B | A’ | B’ | A . B | (A .B)’ | A’ + B’ |
0 0 1 1 | 0 1 0 1 | 1 1 0 0 | 1 0 1 0 | 0 0 0 1 | 1 1 1 0 | 1 1 1 0 |
II. Soal :
1. Give the relationship that represents the dual of the Boolean property A + 1 = 1?
(Note: * = AND, + = OR and ' = NOT)
1. A * 1 = 1
2. A * 0 = 0 (Jawabanya)
3. A + 0 = 0
4. A * A = A
5. A * 1 = 1
2. Give the best definition of a literal?
1. A Boolean variable
2. The complement of a Boolean variable (Jawabannya)
3. 1 or 2
4. A Boolean variable interpreted literally
5. The actual understanding of a Boolean variable
3. Simplify the Boolean expression (A+B+C)(D+E)' + (A+B+C)(D+E) and choose the best answer.
1. A + B + C (Jawabannya)
2. D + E
3. A’B’C’
4. D’E’
5. None of the above
4.Which of the following relationships represents the dual of the Boolean property x + x'y = x + y?
1. x’(x+y’) = x’y’ (Jawabannya)
2 x(x’y) = xy
3. x*x’ + y = xy
4. x’(xy’) = x’y’
5. x(x’ + y) = xy
5.Given the function F(X,Y,Z) = XZ + Z(X'+ XY), the equivalent most simplified Boolean representation for F is:
1. Z + YZ
2. Z + XYZ (Jawabannya)
3. XZ
4. X + YZ
5. None of the above
6. Which of the following Boolean functions is algebraically complete?
1. F = xy (Jawabannya)
2. F = x + y
3. F = x’
4. F = xy +yz
5. F = x + y’
7. Simplification of the Boolean expression (A + B)'(C + D + E)' + (A + B)' yields which of the following results?
1. A + B
2. A’B’ (Jawabannya)
3. C + D + E
4. C’D’E’
5. A’B’C’D’E’
8. Given that F = A'B'+ C'+ D'+ E', which of the following represent the only correct expression for F'?
1. F’= A+B+C+D+E
2. F’= ABCDE
3. F’= AB(C+D+E)
4. F’= AB+C’+D’+E’
5. F’= (A+B)CDE (Jawabannya)
9. An equivalent representation for the Boolean expression A' + 1 is
1. A
2. A’
3. 1 (Jawabannya)
4. 0
10. Simplification of the Boolean expression AB + ABC + ABCD + ABCDE + ABCDEF yields which of the following results?
1. ABCDEF
2.AB (Jawabannya)
3.AB +CD +EF
4. A+B+C+D+E+F