Boolean Algebra: (PHC504)

Boolean Algebra in Digital Electronics

Boolean Algebra is a mathematical system used in digital electronics to analyze and simplify logic expressions. It deals with binary variables and operations, where the variables can take on only two values: 0 (false) and 1 (true). Boolean Algebra operations are based on a set of rules and identities that help simplify complex logic expressions and design efficient digital circuits.

Basic Boolean Operations:

1. AND Operation (·): Represents the logical AND operation. If both inputs are true (1), the output is true; otherwise, it is false (0).
2. OR Operation (+): Represents the logical OR operation. If at least one input is true (1), the output is true; otherwise, it is false (0).
3. NOT Operation ('): Represents the logical NOT operation. It inverses the input; true becomes false, and false becomes true.

Boolean Identities:**

1. Identity Law:

• AND: A · 1 = A
• OR: A + 0 = A

1. Null Law:

• AND: A · 0 = 0
• OR: A + 1 = 1

1. Domination Law:

• AND: A · A = A
• OR: A + A = A

1. Complement Law:

• AND: A · A = 0
• OR: A + A = 1

1. Double Negation Law:

• AND: (A̿) = A
• OR: (A̿) = A

1. Commutative Law:

• AND: A · B = B · A
• OR: A + B = B + A

1. Associative Law:

• AND: (A · B) · C = A · (B · C)
• OR: (A + B) + C = A + (B + C)

1. Distributive Law:

• AND: A · (B + C) = (A · B) + (A · C)
• OR: A + (B · C) = (A + B) · (A + C)

1. De Morgan's Law:

• AND: (A + B) = A · B
• OR: (A · B) = A + B

Examples:

Expression: Y = A + (A · B)

**Simplified:** A + B (Using the Absorption Law)

Expression: Y = A · (A + B)

**Simplified:** A (Using the Absorption Law)

Expression: Y = A · (B + C) + A · B

**Simplified:** A · C + A · B (Using the Distribution Law)

Expression: Y = (A + B) · (A + C) + B

**Simplified:** B + A · C (Using the Distribution Law)

Expression: Y = A + A

**Simplified:** 1 (Using the Complement Law)

Expression: Y = A · 0

**Simplified:** 0 (Using the Null Law)

Expression: Y = A · B + A · B

**Simplified:** B (Using the Absorption Law)

Expression: Y = A · (B + C) · B

**Simplified:** 0 (Using the Domination Law)

Expression: Y = (A + B) · (A + B)

**Simplified:** A · B (Using De Morgan's Law)

Expression: Y = (A · B) + (C + D)

**Simplified:** A + B + C (Using De Morgan's Law)

These examples demonstrate how Boolean identities can be used to simplify complex expressions in Boolean Algebra, leading to more efficient digital circuit designs and logical reasoning.