PowerPoint PresentationBoolean algebraBoolean operatorsMore definitionsLogic symbolsTheoremsMore TheoremsDualityN-variable TheoremsDeMorgan Symbol EquivalenceLikewise for ORDeMorgan SymbolsEven more definitionsMintermMaxtermTruth table vs. minterms & maxtermsCombinational analysisSignal expressionsNew circuit, same function“Add out” logic functionShortcut: Symbol substitutionDifferent circuit, same functionPracticeSlide 24Next ClassEE365Adv. Digital Circuit DesignClarkson UniversityLecture #2Boolean Laws and MethodsBoolean algebra•a.k.a. “switching algebra”–deals with boolean values -- 0, 1•Positive-logic convention–analog voltages LOW, HIGH --> 0, 1•Signal values denoted by variables(X, Y, FRED, etc.)Rissacher EE365Lect #2Boolean operators•Complement: X (opposite of X)•AND: X  Y•OR: X + Ybinary operators, describedfunctionally by truth table.Rissacher EE365Lect #2More 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Rissacher EE365Lect #2Logic symbolsRissacher EE365Lect #2TheoremsRissacher EE365Lect #2More TheoremsRissacher EE365Lect #2Duality•Swap 0 & 1, AND & OR–Result: Theorems still true–Note duals in previous 2 tables (e.g. T6 and T6’)–Example:Rissacher EE365Lect #2N-variable Theorems•Most important: DeMorgan theoremsRissacher EE365Lect #2DeMorgan Symbol EquivalenceRissacher EE365Lect #2Likewise for ORRissacher EE365Lect #2DeMorgan SymbolsRissacher EE365Lect #2Even more definitions•Product term –W•X’•Y•Sum-of-products expression–(W•X’•Y)+(X•Z)+(W’•X’•Y’)•Sum term–A+B’+C•Product-of-sums expression–(A+B’+C)•(D’+A’)•(D+B+C)•Normal term–No variable appears more than once–(W•X’•Y)+(A•Z)+(B’•C’)•Minterm (n variables)•Maxterm (n variables)Rissacher EE365Lect #2Minterm•An n-variable minterm is a normal product term with n literals•There are 2n possibilities•3-variable example: X’•Y’•Z or ΣX,Y,Z(1) •A minterm is a product term that is 1 in exactly one row of the truth table:Rissacher EE365Lect #2X Y Z F0 0 0 00 0 1 10 1 0 00 1 1 01 0 0 01 0 1 01 1 0 01 1 1 0new notationMaxterm•An n-variable maxterm is a normal sum term with n literals•There are 2n possibilities•3-variable example: X’+Y’+Z or ЛX,Y,Z(6) •A maxterm is a sum term that is 0 in exactly one row of the truth table:Rissacher EE365Lect #2X Y Z F0 0 0 10 0 1 10 1 0 10 1 1 11 0 0 11 0 1 11 1 0 01 1 1 1new notationTruth table vs. minterms & maxtermsRissacher EE365Lect #2Combinational analysisRissacher EE365Lect #2Signal expressions•Multiply out:F = ((X + Y)  Z) + (X  Y  Z) = (X  Z) + (Y  Z) + (X  Y  Z)Rissacher EE365Lect #2New circuit, same functionRissacher EE365Lect #2F = ((X + Y)  Z) + (X  Y  Z) = (X  Z) + (Y  Z) + (X  Y  Z)“Add out” logic function•Circuit:Rissacher EE365Lect #2Shortcut: Symbol substitutionRissacher EE365Lect #2Different circuit, same functionRissacher EE365Lect #2PracticeRissacher EE365Lect #2Convert the following function into a POS:F = ((X + Z) • Y) + (X’ • Z’ • Y’)Convert the following function into a POS:F = ((X + Z) • Y) + (X’ • Z’ • Y’)F = (X + Z + X’) • (X + Z + Z’) • (X + Z + Y’) • (Y + X’) • (Y + Z’) • (Y + Y’)F = 1 • 1 • (X + Z + Y’) • (Y + X’) • (Y + Z’) • 1 F = (X + Z + Y’) • (Y + X’) • (Y + Z’) PracticeRissacher EE365Lect #2Next ClassRissacher EE365Lect #2•Building Combination Circuits•Minimization•Karnaugh