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Chapter 2 Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu http://www.ee.unlv.edu/~b1morris/cpe100/ CPE100: Digital Logic Design I Combinational Logic Design Chapter 2 <1> Chapter 2 :: Topics Introduction Boolean

  1. T5: Complements Theorem • B B = 0 • B + B = 1 B = 0 B B = 1 B Chapter 2 <50>

  2. Recap: Basic Boolean Theorems Chapter 2 <51>

  3. Chapter 2.3.3 Theorems of Several Variables

  4. Boolean Theorems of Several Vars Number Theorem Name T6 B•C = C • B Commutativity T7 (B•C) • D = B • (C • D) Associativity T8 B • (C + D) = (B • C) + (B • D) Distributivity T9 B• (B+C) = B Covering T10 (B•C) + (B•C) = B Combining T11 B•C + (B•D) + (C•D) = Consensus B • C + B • D Chapter 2 <53>

  5. Boolean Theorems of Several Vars Number Theorem Name T6 B•C = C • B Commutativity T7 (B•C) • D = B • (C • D) Associativity T8 B • (C + D) = (B • C) + (B • D) Distributivity T9 B• (B+C) = B Covering T10 (B•C) + (B•C) = B Combining T11 B•C + (B•D) + (C•D) = Consensus B • C + B • D How do we prove these are true? Chapter 2 <54>

  6. How to Prove Boolean Relation • Method 1: Perfect induction • Method 2: Use other theorems and axioms to simplify the equation • Make one side of the equation look like the other Chapter 2 <55>

  7. Proof by Perfect Induction • Also called: proof by exhaustion • Check every possible input value • If two expressions produce the same value for every possible input combination, the expressions are equal Chapter 2 <56>

  8. Example: Proof by Perfect Induction Number Theorem Name T6 B•C = C • B Commutativity B C BC CB 0 0 0 1 1 0 1 1 Chapter 2 <57>

  9. Example: Proof by Perfect Induction Number Theorem Name T6 B•C = C • B Commutativity B C BC CB 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 Chapter 2 <58>

  10. Boolean Theorems of Several Vars Number Theorem Name T6 B•C = C • B Commutativity T7 (B•C) • D = B • (C • D) Associativity T8 B • (C + D) = (B • C) + (B • D) Distributivity T9 B• (B+C) = B Covering T10 (B•C) + (B•C) = B Combining T11 B•C + (B•D) + (C•D) = Consensus B • C + B • D Chapter 2 <59>

  11. T7: Associativity Number Theorem Name T7 (B•C) • D = B • (C • D) Associativity Chapter 2 <60>

  12. T8: Distributivity Number Theorem Name T8 B • (C + D) = (B • C) + (B • D) Distributivity Chapter 2 <61>

  13. T9: Covering Number Theorem Name T9 B• (B+C) = B Covering Chapter 2 <62>

  14. T9: Covering Number Theorem Name T9 B• (B+C) = B Covering Prove true by: • Method 1: Perfect induction • Method 2: Using other theorems and axioms Chapter 2 <63>

  15. T9: Covering Number Theorem Name T9 B• (B+C) = B Covering Method 1: Perfect Induction B C (B+C) B(B+C) 0 0 0 1 1 0 1 1 Chapter 2 <64>

  16. T9: Covering Number Theorem Name T9 B• (B+C) = B Covering Method 1: Perfect Induction B C (B+C) B(B+C) 0 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 Chapter 2 <65>

  17. T9: Covering Number Theorem Name T9 B• (B+C) = B Covering Method 2: Prove true using other axioms and theorems. Chapter 2 <66>

  18. T9: Covering Number Theorem Name T9 B• (B+C) = B Covering Method 2: Prove true using other axioms and theorems. B •(B+C) = B•B + B•C T8: Distributivity = B + B•C T3: Idempotency = B•(1 + C) T8: Distributivity = B•( 1 ) T2: Null element = B T1: Identity Chapter 2 <67>

  19. T10: Combining Number Theorem Name T10 (B•C) + (B•C) = B Combining Prove true using other axioms and theorems: Chapter 2 <68>

  20. T10: Combining Number Theorem Name T10 (B•C) + (B•C) = B Combining Prove true using other axioms and theorems: B•C + B• C = B•(C+C) T8: Distributivity = B •( 1 ) T5 ’ : Complements = B T1: Identity Chapter 2 <69>

  21. T11: Consensus Number Theorem Name T11 (B•C) + (B•D) + (C•D) = Consensus (B • C) + B • D Prove true using (1) perfect induction or (2) other axioms and theorems. Chapter 2 <70>

  22. Recap: Boolean Thms of Several Vars Number Theorem Name T6 B•C = C • B Commutativity T7 (B•C) • D = B • (C • D) Associativity T8 B • (C + D) = (B • C) + (B • D) Distributivity T9 B• (B+C) = B Covering T10 (B•C) + (B•C) = B Combining T11 B•C + (B•D) + (C•D) = Consensus B • C + B • D Chapter 2 <71>

  23. Boolean Thms of Several Vars: Duals # Theorem Dual Name T6 B•C = C • B B+C = C+B Commutativity T7 (B•C) • D = B • (C • D) (B + C) + D = B + (C + D) Associativity T8 B • (C + D) = (B • C) + (B • D) B + (C • D) = (B+C) (B+D) Distributivity T9 B • (B+C) = B B + (B•C) = B Covering T10 (B•C) + (B•C) = B (B+C) • (B+C) = B Combining T11 (B•C) + (B•D) + (C•D) = (B+C) • (B+D) • (C+D) = Consensus (B • C) + (B • D) (B+C) • (B+D) Dual: Replace: • with + 0 with 1 Chapter 2 <72>

  24. Boolean Thms of Several Vars: Duals # Theorem Dual Name T6 B•C = C • B B+C = C+B Commutativity T7 (B•C) • D = B • (C • D) (B + C) + D = B + (C + D) Associativity T8 B • (C + D) = (B • C) + (B • D) B + (C • D) = (B+C) (B+D) Distributivity T9 B • (B+C) = B B + (B•C) = B Covering T10 (B•C) + (B•C) = B (B+C) • (B+C) = B Combining T11 (B•C) + (B•D) + (C•D) = (B+C) • (B+D) • (C+D) = Consensus (B • C) + (B • D) (B+C) • (B+D) Dual: Replace: • with + 0 with 1 Warning: T8 ’ differs from traditional algebra: OR (+) distributes over AND (•) Chapter 2 <73>

  25. Boolean Thms of Several Vars: Duals # Theorem Dual Name T6 B•C = C • B B+C = C+B Commutativity T7 (B•C) • D = B • (C • D) (B + C) + D = B + (C + D) Associativity T8 B • (C + D) = (B • C) + (B • D) B + (C • D) = (B+C) (B+D) Distributivity T9 B • (B+C) = B B + (B•C) = B Covering T10 (B•C) + (B•C) = B (B+C) • (B+C) = B Combining T11 (B•C) + (B•D) + (C•D) = (B+C) • (B+D) • (C+D) = Consensus (B • C) + (B • D) (B+C) • (B+D) Axioms and theorems are useful for simplifying equations. Chapter 2 <74>

  26. Chapter 2.3.5 Simplifying Equations

  27. Simplifying an Equation • Reducing an equation to the fewest number of implicants, where each implicant has the fewest literals Chapter 2 <76>

  28. Simplifying an Equation • Reducing an equation to the fewest number of implicants, where each implicant has the fewest literals Recall:  Implicant: product of literals ABC , AC , BC  Literal: variable or its complement A , A , B , B , C , C Chapter 2 <77>

  29. Simplifying an Equation • Reducing an equation to the fewest number of implicants, where each implicant has the fewest literals Recall:  Implicant: product of literals ABC , AC , BC  Literal: variable or its complement A , A , B , B , C , C • Also called: minimizing the equation Chapter 2 <78>

  30. Simplification methods • Distributivity (T8, T8 ’ ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) • 𝑄 𝐵 Minterm Covering (T9 ’ ) 𝑄 ҧ ത 𝐵 0 0 A + AP = A ത 𝑄𝐵 0 1 • Combining (T10) 𝑄 ҧ 1 0 𝐵 PA + PA = P 1 1 𝑄𝐵 Chapter 2 <79>

  31. Simplification methods • Distributivity (T8, T8 ’ ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) • 𝑄 𝐵 Minterm Covering (T9 ’ ) 𝑄 ҧ ത 𝐵 0 0 A + AP = A ത 𝑄𝐵 0 1 • Combining (T10) 𝑄 ҧ 1 0 𝐵 PA + PA = P 1 1 𝑄𝐵 • Expansion P = PA + PA A = A + AP • Duplication A = A + A Chapter 2 <80>

  32. Simplification methods • Distributivity (T8, T8 ’ ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) • Covering (T9 ’ ) A + AP = A • Combining (T10) PA + PA = P • Expansion P = PA + PA A = A + AP • Duplication A = A + A • A combination of Combining/Covering PA + A = P + A Chapter 2 <81>

  33. Simplification methods • A combination of Combining/Covering PA + A = P + A PA + A = PA + ( A + AP) T9 ’ Covering Proof: = PA + PA + A T6 Commutativity = P(A + A) + A T8 Distributivity T5 ’ Complements = P (1) + A T1 Identity = P + A Chapter 2 <82>

  34. Recap: Boolean Thms of Several Vars # Theorem Dual Name T6 B•C = C • B B+C = C+B Commutativity T7 (B•C) • D = B • (C • D) (B + C) + D = B + (C + D) Associativity T8 B • (C + D) = (B • C) + (B • D) B + (C • D) = (B+C) (B+D) Distributivity T9 B • (B+C) = B B + (B•C) = B Covering T10 (B•C) + (B•C) = B (B+C) • (B+C) = B Combining T11 (B•C) + (B•D) + (C•D) = (B+C) • (B+D) • (C+D) = Consensus (B • C) + (B • D) (B+C) • (B+D) Chapter 2 <83>

  35. T11: Consensus Number Theorem Name T11 (B•C) + (B•D) + (C•D) = Consensus (B • C) + B • D Prove using other theorems and axioms: Chapter 2 <84>

  36. T11: Consensus Number Theorem Name T11 (B•C) + (B•D) + (C•D) = Consensus (B • C) + B • D Prove using other theorems and axioms: B•C + B•D + C•D T10: Combining = BC + BD + (CDB+CDB) = BC + BD + BCD+BCD T6: Commutativity = BC + BCD + BD + BCD T6: Commutativity T7: Associativity = (BC + BCD) + (BD + BCD) T9 ’ : Covering = BC + BD Chapter 2 <85>

  37. Simplification methods • Distributivity (T8, T8 ’ ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) • Covering (T9 ’ ) A + AP = A • Combining (T10) PA + PA = P • Expansion P = PA + PA A = A + AP • Duplication A = A + A • A combination of Combining/Covering PA + A = P + A Chapter 2 <86>

  38. Simplifying Boolean Equations Example 1: Y = AB + AB Chapter 2 <87>

  39. Simplifying Boolean Equations Example 1: Y = AB + AB Y = A T10: Combining or = A ( B + B ) T8: Distributivity = A (1) T5’: Complements = A T1: Identity Chapter 2 <88>

  40. Simplification methods • Distributivity (T8, T8 ’ ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) • Covering (T9 ’ ) A + AP = A • Combining (T10) PA + PA = P • Expansion P = PA + PA A = A + AP • Duplication A = A + A • A combination of Combining/Covering PA + A = P + A Chapter 2 <89>

  41. Simplifying Boolean Equations Example 2: Y = A ( AB + ABC) Chapter 2 <90>

  42. Simplifying Boolean Equations Example 2: Y = A ( AB + ABC) = A ( AB( 1 + C )) T8: Distributivity = A ( AB (1)) T2’: Null Element = A ( AB ) T1: Identity = ( AA ) B T7: Associativity = AB T3: Idempotency Chapter 2 <91>

  43. Simplification methods • Distributivity (T8, T8 ’ ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) • Covering (T9 ’ ) A + AP = A • Combining (T10) PA + PA = P • Expansion P = PA + PA A = A + AP • Duplication A = A + A • A combination of Combining/Covering PA + A = P + A Chapter 2 <92>

  44. Simplifying Boolean Equations Example 3: Y = A’BC + A’ Recall: A’ = A Chapter 2 <93>

  45. Simplifying Boolean Equations Example 3: Y = A’BC + A’ Recall: A’ = A = A’ T9’ Covering: X + XY = X or = A’(BC + 1) T8: Distributivity = A’(1) T2’: Null Element = A’ T1: Identity Chapter 2 <94>

  46. Simplification methods • Distributivity (T8, T8 ’ ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) • Covering (T9 ’ ) A + AP = A • Combining (T10) PA + PA = P • Expansion P = PA + PA A = A + AP • Duplication A = A + A • A combination of Combining/Covering PA + A = P + A Chapter 2 <95>

  47. Simplifying Boolean Equations Example 4: Y = AB’C + ABC + A’BC Chapter 2 <96>

  48. Simplifying Boolean Equations Example 4: Y = AB’C + ABC + A’BC = AB’C + ABC + ABC + A’BC T3’: Idempotency = (AB’C+ABC) + (ABC+A’BC) T7’: Associativity = AC + BC T10: Combining Chapter 2 <97>

  49. Simplification methods • Distributivity (T8, T8 ’ ) B (C+D) = BC + BD B + CD = (B+ C)(B+D) • Covering (T9 ’ ) A + AP = A • Combining (T10) PA + PA = P • Expansion P = PA + PA A = A + AP • Duplication A = A + A • A combination of Combining/Covering PA + A = P + A Chapter 2 <98>

  50. Simplifying Boolean Equations Example 5: Y = AB + BC +B’D’ + AC’D’ Chapter 2 <99>

  51. Simplifying Boolean Equations Example 5: Y = AB + BC +B’D’ + AC’D’ Method 1: Y = AB + BC + B’D’ + (ABC’D’ + AB’C’D’) T10: Combining = (AB + ABC’D’) + BC + (B’D’ + AB’C’D’) T6: Commutativity T7: Associativity = AB + BC + B’D’ T9: Covering Chapter 2 <100>

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