Unit 4: Boolean Algebra and Logic Gates

1. Binary Number Systems and Conversions

Digital systems operate using Binary Numbers (base-2), consisting of only 0 and 1.

• Binary to Decimal: Each binary digit (bit) is multiplied by 2 raised to the power of its position (starting from 0 at the right).
• Decimal to Binary: Repeated division of the decimal number by 2, recording the remainders.
• BCD (Binary Coded Decimal): A system where each decimal digit is represented by a 4-bit binary code.

2. Basic Logic Gates: AND, OR, NOT

Logic gates are the building blocks of digital circuits, often realized using diodes and transistors.

• AND Gate: Output is HIGH (1) only if all inputs are HIGH.
• OR Gate: Output is HIGH (1) if at least one input is HIGH.
• NOT Gate: Also known as an inverter; it outputs the opposite of the input.
• XOR Gate: Output is HIGH only if the inputs are different.

3. Universal Gates: NAND and NOR

NAND and NOR gates are called Universal Gates because any boolean function can be implemented using only one type of these gates.

• NAND: NOT-AND; output is LOW only when all inputs are HIGH.
• NOR: NOT-OR; output is HIGH only when all inputs are LOW.

4. De Morgan's Theorems and Boolean Laws

These theorems are essential for simplifying complex boolean expressions.

First Theorem: (A + B)' = A' · B' (The complement of a sum is equal to the product of the complements).
Second Theorem: (A · B)' = A' + B' (The complement of a product is equal to the sum of the complements).

Important Boolean Laws:

• Commutative: A + B = B + A; A · B = B · A.
• Associative: A + (B + C) = (A + B) + C.
• Distributive: A · (B + C) = (A · B) + (A · C).
• Identity: A + 0 = A; A · 1 = A.

5. Simplification of Logic Circuits

Simplification involves using Boolean laws and theorems to reduce the number of gates required for a specific truth table. This reduces cost, space, and power consumption in hardware design.

6. Canonical Forms: SOP and POS

Any logic expression can be written in two standard forms based on its truth table.

• Minterms: Product terms that represent an output of 1.
• Maxterms: Sum terms that represent an output of 0.
• SOP (Sum of Products): A group of ANDed terms (minterms) ORed together.
• POS (Product of Sums): A group of ORed terms (maxterms) ANDed together.

7. Karnaugh Map (K-Map)

The Karnaugh Map is a graphical tool used to simplify Boolean expressions without using complex laws. It organizes minterms into a grid where adjacent cells differ by only one bit, allowing for easy identification of redundant variables.