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Simplifying Boolean Expressions

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Dive into the world of Boolean expressions with a comprehensive overview of the laws and operators used to simplify them. This guide covers the fundamental properties such as commutative, associative, and distributive laws, along with tautology and absorption laws. Learn how to apply these principles, utilize truth tables for verification, and explore De Morgan’s laws for effective simplification. Whether you’re a student or a professional, mastering these concepts will enhance your understanding of digital logic design and computational applications.

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Simplifying Boolean Expressions

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  1. Simplifying Boolean Expressions

  2. Boolean Operators (T/F)

  3. Boolean Operators (1/0)

  4. Boolean Operators Symbols

  5. Simplifying Boolean Expressions • Commutative laws A + B = B + A A · B = B · A • Associative laws A + (B + C) = (A + B) + C A · (B · C) = (A · B) · C • Distributive laws A · (B + C) = A · B + A · C A + (B · C) = (A + B) · (A + C)

  6. Simplifying Boolean Expressions • Tautology laws A · A = A A + A = A A + ~A = 1 A · ~A = 0 • Absorption Law A + (A · B) = A A · (A + B) = A

  7. Simplifying Boolean Expressions • Identities 0 · A = 0 0 + A = A A + 1 = 1 1 · A = A A = A • Complement A + ~A · B = A + B

  8. Examples • A + A + A + A = A Using the Tautology law

  9. A Bigger Example Simplify ~A · B + A · ~B + ~A · ~B ~A · B + A · ~B + ~A · ~B ~A · B + (A · ~B + ~A · ~B)  Associative ~A · B + (~B · (A + ~A))  Distributive ~A · B + ~B & Tautology ~A + ~B  Complement Verify with a truth table!

  10. Practice • Show that A + B · C = (A + B) · (A + C) is true using a truth table.

  11. Practice • Show that A + ~A · B = A + B

  12. Practice Simplification • Simplify A + AB + ~B and verify with a truth table

  13. De Morgan’s Laws ~(A · B) = ~A + ~B ~A · ~B = ~(A+B) • Take a term ~A · ~B • NOT the individual members of the term A · B • Change the operator i.e. · to +, or + to · A + B • NOT the entire term ~(A+B)

  14. De Morgan’s Law Example f = ~A · ~B + (~A + ~B) = ~~( ~A · ~B + (~A + ~B) )  NOT NOT = ~( (A + B) · ~(~A + ~B) )  De Morgan’s = ~( (A + B) · (A·B) )  De Morgan’s = ~( A·(A·B) + B·(A·B) )  Distributive = ~( A·B + A·B )  Tautology = ~(A·B)  Tautology

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