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Simplifying Boolean Expressions

Simplifying Boolean Expressions. Boolean Operators (T/F). Boolean Operators (1/0). Boolean Operators Symbols. 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

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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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