Boolean Algebra Laws

Boolean algebra is the mathematics of logic, operating on variables that can only be TRUE (1) or FALSE (0). Just as regular algebra has rules for simplifying expressions, Boolean algebra has its own set of laws and theorems. Mastering these rules allows you to simplify complex logic expressions, reducing the number of gates needed in a circuit.

Objectives

• State and apply all fundamental Boolean algebra laws
• Prove and use De Morgan's two theorems
• Apply the absorption law to simplify expressions
• Use the consensus theorem for advanced simplification
• Understand and apply the duality principle
• Simplify multi-variable Boolean expressions step by step

Key Takeaways

• Boolean algebra uses only 0 and 1, with AND (·), OR (+), and NOT (bar) operations
• De Morgan's is the most important theorem: "break the bar, change the sign"
• Absorption law (A + AB = A) eliminates redundant terms
• Consensus theorem removes terms formed from the remaining variables of complementary pairs
• Every theorem has a dual (swap AND↔OR, 0↔1)
• Always verify simplifications with a truth table if unsure

Applications

• Circuit Optimization: Fewer gates = less cost, power, and delay.
• Software Logic: Simplifying conditional expressions in programming.
• Digital Design: Foundation for Karnaugh maps and automated synthesis tools.
• Verification: Proving two circuits are functionally equivalent.

Practice Problems

Problem 1: Simplify: A·B + A·B' + A'·B

Problem 2: Apply De Morgan's to simplify: NOT(A·B + C)

Problem 3: Simplify: X·Y + X'·Z + Y·Z

Problem 4: Find the dual of: A·(B+C) = A·B + A·C

Problem 5: Simplify: (A+B)·(A+C)·(B+C)

Problem 6: Prove by truth table that A + A'·B = A + B