Boolean Expression Minimization

Jul 23, 2024

Boolean Expression Minimization

Introduction

  • Presenter: Jayashree Gupta
  • Topic: Minimizing a Boolean expression

Example Boolean Expression

  • Expression: (\overline{A}B + A + A\overline{B})
  • Consists of 3 terms: (\overline{A}B), (A), (A\overline{B})
  • Goal: Simplify or minimize the expression

Methods of Minimization

  1. Using the Laws of Boolean Algebra
  2. Using Karnaugh Map (K-map)

Frequently Used Boolean Algebra Laws

  1. Identity Laws:
    • (A + \overline{A} = 1)
    • (A \cdot \overline{A} = 0)
  2. Annulment Laws:
    • (1 + A = 1)
    • (0 + A = A)
    • (1 \cdot A = A)
    • (0 \cdot A = 0)
  3. De Morgan’s Laws:
    • ((A + B)^{'} = \overline{A} \cdot \overline{B})
    • ((A \cdot B)^' = \overline{A} + \overline{B})
  4. Distributive Laws:
    • (A \cdot (B + C) = (A \cdot B) + (A \cdot C))
    • (A + (B \cdot C) = (A + B) \cdot (A + C))
  5. Redundant Literal Rule:
    • Example: (A + \overline{A}B = A + B)

Minimization Techniques

Example 1

  • Expression: (BC + \overline{B}C)
  • Simplification:
    • Take common (C)
    • (C(A + B + \overline{B}) = C(A + 1) = A + C)

Example 2

  • Expression: ((\overline{A}B + AB))
  • Steps:
    • Apply distributive law and redundant literal rule
    • Simplified to (A + B)

Example 3

  • Expression: (A + AB\overline{C} + \overline{A}B)
  • Simplification:
    • Combine terms where possible
    • Apply laws systematically
    • Simplified to (A + B)

Advanced Examples

Example 4

  • Expression: (AB + AC\overline{B}C \cdot AB)
  • Steps:
    • Apply De Morgan's law and distributive properties
    • Simplification may include multiple layers of applying laws
    • Final minimized expression Various complex steps involved

Example 5

  • Expression: ((AB + ABC) + A(BC + A\overline{B}C))
  • Strategies:
    • Apply redundant literal rule, De Morgan's law
    • Identify and cancel out contradictory terms

Conclusion

  • Simplification of Boolean expressions involves the systematic application of the Boolean algebra laws.
  • Common laws include Identity laws, De Morgan’s laws, Distributive laws, and the Redundant Literal Rule.
  • Practicing multiple examples helps in mastering the minimization process.