1.2 Boolean Algebra
Boolean Algebra is a branch of mathematics that deals with logical operations on binary variables. These variables take only two values: 0 (False) and 1 (True).
It is also known as Binary Algebra, Two-Valued Logic, or Logical Algebra. Boolean Algebra was introduced by George Boole in 1847.
This concept is the foundation of digital electronics, VLSI design, and programming.
Logic Levels
1. Positive Logic
- Logic 0 = False, 0V, Open Switch, OFF
- Logic 1 = True, +5V, Closed Switch, ON
2. Negative Logic
- Logic 0 = True, +5V, Closed Switch, ON
- Logic 1 = False, 0V, Open Switch, OFF
Boolean Algebra vs Normal Algebra
| Feature | Boolean Algebra | Normal Algebra |
|---|---|---|
| Values | 0 and 1 | Decimal Numbers |
| Operations | AND, OR, NOT | +, −, ×, ÷ |
| Nature | Logical | Numerical |
Binary: 1 + 1 = 10
Boolean: 1 + 1 = 1
Boolean: 1 + 1 = 1
1.2.1 Rules and Properties
- Boolean variables take only 0 and 1
- Operations: NOT, OR, AND
- Expressions are called Boolean Expressions
- NOT: Y = A̅
- OR: Y = A + B
- AND: Y = A . B
Truth Tables
Table 1.1: Complement
| A | Y |
|---|---|
| 0 | 1 |
| 1 | 0 |
Table 1.2: OR Operation
| A | B | Y |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Table 1.3: AND Operation
| A | B | Y |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
1.2.2 Boolean Laws
Commutative Law
A . B = B . A
A + B = B + A
Associative Law
A(B.C) = (A.B).C
A+(B+C) = (A+B)+C
Distributive Law
A(B+C) = AB + AC
A + BC = (A+B)(A+C)
AND Laws
A.0 = 0
A.1 = A
A.A = A
A.A̅ = 0
OR Laws
A+0 = A
A+1 = 1
A+A = A
A+A̅ = 1
Complement Laws
0̅ = 1
1̅ = 0
(A̅)̅ = A
Absorption Law
A(A+B)=A
A+AB=A
A+A̅B=A+B
De Morgan’s Laws
(A + B + C)̅ = A̅ . B̅ . C̅
Table 1.4: De-Morgan's First Law
(A . B . C)̅ = A̅ + B̅ + C̅
Table 1.5: De-Morgan's Second Law
Operator Precedence
| Operation | Priority |
|---|---|
| Parenthesis | 1 |
| NOT | 2 |
| AND | 3 |
| OR | 4 |
Example
Y = (A(C + B̅D̅) + BC̅)̅