# Harold’s Boolean Algebra Cheat Sheet Equivalent .

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Harold’s Boolean AlgebraCheat Sheet12 September 2021Boolean AlgebraBooleanExpressionLaw or RuleEquivalentCircuit𝐴 1 1Annulment(OR)A in parallel with closed “CLOSED”𝐴 0 0Annulment(AND)A in series with open “OPEN”𝐴 0 𝐴Identity(OR)A in parallel with open “A”𝐴 1 𝐴Identity(AND)A in series with closed “A”𝐴 𝐴 𝐴Idempotent(OR)A in parallel with A “A”𝐴𝐴 𝐴Idempotent(AND)A in series with A “A”̅̅̅̅̅̅̅( 𝐴̅ ) 𝐴Double NegationNOT NOT A (double negative) “A”𝐴 𝐴 1Complement(OR)A in parallel with NOT A “CLOSED”𝐴𝐴 0Complement(AND)A in series with NOT A “OPEN”𝐴 𝐵 𝐵 𝐴Commutative(OR)A in parallel with B B in parallelwith A𝐴𝐵 𝐵𝐴Commutative(AND)A in series with B B in series with A𝐴(𝐵 𝐶) 𝐴𝐵 𝐴𝐶𝐴 𝐵𝐶 (𝐴 𝐵)(𝐴 𝐶)Distributative(OR)Distributative(AND)Permits the multiplying or factoringout of an expressionCopyright 2021 by Harold Toomey, WyzAnt TutorDescription1

𝐴 (𝐵 𝐶) (𝐴 𝐵) 𝐶 𝐴 𝐵 𝐶𝐴(𝐵𝐶) (𝐴𝐵)𝐶 𝐴𝐵𝐶𝐴 (𝐴𝐵) 𝐴𝐴(𝐴 𝐵) 𝐴𝐴 𝐴𝐵 𝐴 𝐵(𝐴 𝐵) 𝐴 𝐵𝐴𝐵 𝐴 𝐵Associative(OR)Associative(AND)Allows the removal of brackets froman expression and regrouping of thevariablesAbsorptive(OR)Absorptive(AND)Enables a reduction in a complicatedexpression to a simpler one byabsorbing like termsAbsorptive(Derived)Reduces a complicated expression toa simpler one by absorbingcompliment termDe Morgan’s Theorem(NOR)De Morgan’s Theorem(NAND)Invert and replace OR with ANDInvert and replace AND with ORSource: https://www.electronics-tutorials.ws/boolean/bool 6.htmlCopyright 2021 by Harold Toomey, WyzAnt Tutor2

Boolean Logic NDNORXORXNORNotation1TTrue0F FalseA, B, C, DW, X, Y, ZA BABA. BA BA BA BA BA BA BAA A′ A A!AABA BA B*A BA BA BA BA BAB ABA BA BAB ABGateDescriptionOn, Tautology, High voltage (typically 5V)GND Copyright 2021 by Harold Toomey, WyzAnt TutorOff, Contradiction, Low voltage(typically 0V)Line, Wire, Connects toLine, Wire, Connects fromAND, BUT, Multiply, Conjunction,IntersectionInclusive-OR, Add, Disjunction, UnionNOT, Invert, Negation, Change,DifferenceNot ANDNot ORExclusive-OR, Both A and B aredifferentExclusive-NOR, Both A and B are thesame3

Boolean Logic Truth TablesInputsAB00110101OutputsANDNANDOR NORXOR XNOR NOT𝐴VCC1GND011100111100001101001A 1A 1A 0A 011110000 0001Blank Truth XYZCopyright 2021 by Harold Toomey, WyzAnt Tutor4

Karnaugh Mapping D11101AB00011110AB000111102x2 Group1x4 GroupK-Map Rules1) Circle only 1s (ones) and don’t cares for Sum of Products (SOP), 𝑒. 𝑔. 𝐴 𝐵 𝐶 𝐴𝐵𝐶 𝐴𝐵𝐶.a. Circle only 0s (zeros) and don’t cares for Product of Sums (POS), 𝑒. 𝑔. (𝐴 𝐵)(𝐴 𝐵).b. Don’t cares may be used or ignored.2) No diagonals, only horizontal or vertical connections.3) Group only adjacent cells in groups with powers of 2 (1x1, 1x2, 2x1, 2x2, 2x4, 4x2, 1x4, 4x1).4) Make groups as large as possible.5) Must group all 1s (ones) for SOP or all 0s (zeros) for POS.6) Overlapping is allowed.7) Wrapping around all edges allowed, both top-bottom edges and left-right edges.8) Fewest groups possible (OPTIMAL).9) For each circle, determine which inputs do not contribute to the logic (is both 0 and 1).10) Write down equation as a SOP, 𝑒. 𝑔. 𝐴 𝐵 𝐶 𝐴𝐵𝐶 𝐴𝐵𝐶Copyright 2021 by Harold Toomey, WyzAnt Tutor5

Harold’s Boolean Algebra Cheat Sheet 12 September 2021 Boolean Algebra Boolean Expression Law or Rule Equivalent Circuit Description s s Annulment (OR) A in parallel with closed “CLOSED” r r Annulment (AND) A in series with open “OPEN” r