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Boolean Algebra: Analysis and Combinational Circuit

Explore the concepts of Boolean algebra and combinatorial circuit analysis. Learn about Boolean operators, theorems, duality, DeMorgan's symbols, and more. Understand truth tables, minterms, and maxterms. Gain insights into signal expressions and logic functions.

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Boolean Algebra: Analysis and Combinational Circuit

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  1. CSC205 Jeffrey N. Denenberg Lecture #3 Boolean algebraCombinational-circuit analysis CSC 205

  2. Boolean algebra • a.k.a. “switching algebra” • deals with Boolean values -- 0, 1 • Positive-logic convention • analog voltages LOW, HIGH --> 0, 1 • Negative logic -- seldom used • Signal values denoted by variables(X, Y, FRED, etc.) CSC 205

  3. Boolean operators • Complement: X¢ (opposite of X) • AND: X × Y • OR: X + Y • Huntington Postulates: 1-6 Ch. 2.2Postulates and Theorems: Table 2-1 binary operators, describedfunctionally by truth table. CSC 205

  4. More definitions • Literal: a variable or its complement • X, X¢, FRED¢, CS_L • Expression: literals combined by AND, OR, parentheses, complementation • X+Y • P × Q × R • A + B × C • ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢ • Equation: Variable = expression • P = ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢ CSC 205

  5. Logic symbols CSC 205

  6. Theorems • See Table 2-1 • Proofs by perfect induction CSC 205

  7. More Theorems • Again Table 2-1 CSC 205

  8. Duality • Swap 0 & 1, AND & OR • Result: Theorems still true • Why? • Each axiom (A1-A5) has a dual (A1¢-A5¢) • Counterexample:X + X × Y = X (T9)X × X + Y = X (dual)X + Y = X (T3¢)???????????? X + (X×Y) = X (T9)X× (X + Y) = X (dual)(X× X) + (X× Y) = X (T8)X+ (X× Y) = X (T3¢) parentheses,operator precedence! CSC 205

  9. N-variable Theorems • Prove using finite induction • Most important: DeMorgan theorems CSC 205

  10. DeMorgan Symbol Equivalence CSC 205

  11. Likewise for OR • (be sure to check errata!) CSC 205

  12. DeMorgan Symbols CSC 205

  13. Even more definitions (Sec. 4.1.6) • Product term • Sum-of-products expression • Sum term • Product-of-sums expression • Normal term • Minterm (n variables) • Maxterm (n variables) CSC 205

  14. Truth table vs. minterms & maxterms • Table 2-3 CSC 205

  15. Combinational analysis CSC 205

  16. Signal expressions • Multiply out:F = ((X + Y¢) × Z) + (X¢× Y × Z¢) = (X × Z) + (Y¢× Z) + (X¢× Y × Z¢) CSC 205

  17. New circuit, same function CSC 205

  18. “Add out” logic function • Circuit: CSC 205

  19. Shortcut: Symbol substitution CSC 205

  20. Different circuit, same function CSC 205

  21. Another example CSC 205

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