Transcript Relationship Between Basic Operation of Boolean and Basic Logic Gate

Relationship Between Basic Operation
of Boolean and Basic Logic Gate
• The basic construction of a logical circuit is gates
• Gate is an electronic circuit that emits an output signal as a
result of a simple Boolean operation on its inputs
• Logical function is presented through the combination of
gates
• The basic gates used in digital logic is the same as the basic
Boolean algebra operations (e.g., AND, OR, NOT,…)
• The package Truth Tables and Boolean Algebra set out the basic
principles of logic.
Name
Graphic Symbol
AND
A
B
F
OR
A
B
F
NOT
NAND
NOR
A
A
B
A
B
Boolean Algebra
F =A. B
Or
F = AB
F =A+ B
_
F =A
F
____
F=A.B
Or
F = AB
F
F
_____
F=A+B
Truth Table
A
0
0
1
1
A
0
0
1
1
B
0
1
0
1
B
0
1
0
1
B
0
1
F
0
0
0
1
F
0
1
1
1
F
1
0
A B
0 0
0 1
1 0
1 1
F
0
1
1
1
A B
0 0
0 1
1 0
1 1
F
0
0
0
1
the symbols, algebra signs and the truth table for the gates
Basic Theorems of Boolean Algebra
1. Identity Elements
1.A=A
0+A=A
2. Inverse Elements
A.A=0
A+A=1
3. Idempotent Laws
A+A=A
A.A=A
4. Boundess Laws
A+1=1
A.0=0
5. Distributive Laws
A . (B + C) = A.B + A.C
A + (B . C) = (A+B) . (A+C)
6. Order Exchange Laws
A.B=B.A
A+B=B+A
7. Absorption Laws
A + (A . B) = A
A . (A + B) = A
9. Elimination Laws

A + (A . B) = A + B

A . (A + B) = A . B
8. Associative Laws
A + (B + C) = (A + B) + C
A . (B . C) = (A . B) . C
10. De Morgan Theorem
  
(A + B) = A . B
  
(A . B) = A + B
Exercise 1
• Apply De Morgan theorem to the following
equations:
F =V+A+ L
F =A+ B + C + D
• Verify the following expressions:
S.T + V.W + R.S.T = S.T + V.W
A.B + A.C + B.A = A.B + A.C
Relationship Between Boolean
Function and Logic Circuit
Boolean function  Q = AB + B
= (NOT A AND B) OR B
Logic circuit
A
A
AB
B
B
Q
= AB + B
Relationship Between Boolean
Function and Logic Circuit
• Any Boolean function can be implemented in electronic
form as a network of gates called logic circuit
A
B
A.B = AB
F = AB + C + D
C
D
C+D
G = A . (B + C + D)
A
G = A . (B + C + D)
B
C
D
B+C+D
C+D
Truth Table
A
A
AB
B
B
Produce a truth table from the logic circuit
A
B
A
AB
Q
0
0
0
1
1
1
0
1
0
1
1
0
0
0
0
1
1
0
0
1
Q
= AB + B
Exercise 2
• Build a truth table for the following
Boolean function
G = A . (B + C + D)
Karnaugh Map
• A graphical way of depicting the content of a truth table
where the adjacent expressions differ by only one variable
• For the purposes simplification, the Karnaugh map is a
convenient way of representing a Boolean function of a
small number (up to four) of variables
• The map is an array of 2n squares, representing all possible
combination of values of n binary variables
• Example: 2 variables, A and B
B
B
B
A
AB
AB
A
AB
AB
A
B
A
0
1
1
0
00
01
10
11
4 variables, A, B, C, D  24 = 16 squares
AB
CD
CD
CD
AB
0000
0001
AB
0100
AB
1100
AB
1000
CD
CD
• List combinations in
the order 00, 01, 11, 10
00
01
11
C
AB
AB
AB
AB
0 C
000
010
110
100
1 C
001
011
111
101
C
0
C
1
C
00
AB
000
001
01
AB
010
011
11
AB
110
111
10
AB
100
101
AB
AB
10
How to create Karnaugh Map
Truth Table
A
B
C
F
0
0
0
1
0
0
1
0
0
1
0
0
0
1
1
1
1
0
0
1
1
0
1
1
1
1
0
0
1
1
1
0
1. Place 1 in the corresponding
square
Karnaugh Map
BC
BC
00
A 0
1
A
A 1
1
BC
01
BC
11
1
1
BC
10
Karnaugh Maps to Represent Boolean Functions
AB
00
AB
01
AB
1
11
AB
10
AB
1
F = AB + AB
2. Group the adjacent squares:
Begin grouping square with 2n-1 for n variables
• e.g. 3 variables, A, B, and C
23-1 = 22 = 4
= 21 = 2
= 20 = 1
BC
BC
00
A 0
1
A
1
A 1
BC
BC
01
BC
11
BC
10
1
1
AB
ABC
F = BC + AB + ABC
BC
A
BC
00
BC
11
BC
10
1
A 0
A 1
BC
01
1
1
A
F = A + BC
BC
1
1
3 variables:
23-1 = 22 = 4
22-1 = 21 = 2
21-1 = 20 = 1
4 variables, A, B, C, D  24-1 = 23 = 8 (maximum); 22 = 4;
21 = 2; 20 = 1 (minimum);
AB
CD
00
00
1
01
1
11
1
10
1
F=
01
11
10
1
1
CD + BD + ABC
1
The following diagram illustrates some of the possible pairs of
values for which simplification is possible:
Karnaugh Map
Boolean Function
Logic Circuit
Exercise 3
Transform the following truth table to Karnaugh Map and
find the Boolean function