Unit 2: Boolean Algebra and Logic Gates
1. Basic and Axiomatic Definitions of Boolean Algebra
Boolean Algebra: A branch of algebra that deals with binary variables and logical operations. It is used in computer science, digital circuits, and logic design.
Binary Variables: Can have only two values: 0 (False) and 1 (True).
Axioms of Boolean Algebra:
- Closure: The result of any operation on binary variables is a binary value (0 or 1).
- Identity Law: For AND: \( A ⋅ 1 = A \), For OR: \( A + 0 = A \).
- Complement Law: \( A ⋅ A' = 0 \), \( A + A' = 1 \).
- 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: \( A ⋅ (B + C) = (A ⋅ B) + (A ⋅ C) \), \( A + (B ⋅ C) = (A + B) ⋅ (A + C) \).
2. Basic Theorems and Properties of Boolean Algebra
- Idempotent Law: \( A + A = A \), \( A ⋅ A = A \).
- Null Law: \( A + 1 = 1 \), \( A ⋅ 0 = 0 \).
- Inverse Law: \( A + A' = 1 \), \( A ⋅ A' = 0 \).
- Absorption Law: \( A + (A ⋅ B) = A \), \( A ⋅ (A + B) = A \).
- De Morgan’s Theorems: \( (A ⋅ B)' = A' + B' \), \( (A + B)' = A' ⋅ B' \).
3. Boolean Functions
A Boolean function is an expression formed using binary variables, logical operators (AND, OR, NOT), and constants (0, 1).
Truth Table Example:
| A | B | F(A, B) = A + B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
4. Logic Operations
AND (\( ⋅ \)):
Outputs 1 if all inputs are 1.
| A | B | A ⋅ B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
OR (\( + \)):
Outputs 1 if at least one input is 1.
| A | B | A + B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
5. Logic Gates
AND Gate:
AND
OR Gate:
OR
NOT Gate:
NOT
6. Integrated Circuits
Integrated Circuits (ICs) are compact electronic circuits that integrate multiple components into a single chip.
- Small-Scale Integration (SSI): Contains a few gates.
- Medium-Scale Integration (MSI): Contains 10 to 100 gates.
- Large-Scale Integration (LSI): Contains 100 to 1000 gates.
- Very Large-Scale Integration (VLSI): Contains thousands or millions of gates.
