C.S. Choy21 BOOLEAN ALGEGRA The Mathematics of logic Boolean variables have only two possible values (binary) Operators:. Product+ SumComplement A.B A+B

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C.S. Choy23 BOOLEAN ALGEBRA Other Properties A+AB = A Proof: A+AB = A+B Proof:

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<p>C.S. Choy21 BOOLEAN ALGEGRA The Mathematics of logic Boolean variables have only two possible values (binary) Operators:. Product+ SumComplement A.B A+B A C.S. Choy22 BOOLEAN ALGEBRA Properties Associative (A+B)+C=A+(B+C)=A+B+C (AB)C=A(BC)=ABC Commutative A+B=B+A AB=BA Distributive A(B+C)=AB+AC Others C.S. Choy23 BOOLEAN ALGEBRA Other Properties A+AB = A Proof: A+AB = A+B Proof: C.S. Choy24 DeMORGANS THEOREM The complement of the SUM function is equal to the PRDUCT function of the complements A+B = AB Equivalent AB = A+B Expansion A+B+C = ABC ABC = A+B+C C.S. Choy25 BOOLEAN ALGEBRA Expression Manipulation (A+B+C)(A+B+C) = C.S. Choy26 TRUTH TABLE Tabulate all possible value combinations of an expression Proof of DeMorgans Theorem A+B = AB ABA+B ABAB C.S. Choy27 LOGIC GATES Building blocks of digital circuits AND Gate Output = AB ABoutput C.S. Choy28 LOGIC GATES OR Gate Output = A + B ABoutput C.S. Choy29 LOGIC GATES Inverter output = A Aoutput 01 10 C.S. Choy30 COMPLETE SET OF OPERATIONS OR, AND and INVERTER together form a complete set because any boolean function can be constructed from a combination of these three gates C.S. Choy31 OTHER KINDS OF GATE NAND Itself a complete set NOR Itself a complete set C.S. Choy32 OTHER KINDS OF GATE Exclusive-OR Gate This is useful as it is functionally equivalent to binary addition XOR = AB + AB = A + B Properties: CommutativeA + B = B + A Associative(A + B) + C = A + (B + C) DistributiveA(B + C) = AB + AC ABA + B C.S. Choy33 EXPRESSION OF DE-MORGANS THEOREM IN TERMS OF LOGIC GATES A + B = AB C.S. Choy34 DESIGN PROCESS F = ABC ABcF The term ABC can be written directly from the truth table as it corresponds with the binary pattern 111 C.S. Choy35 DESIGN PROCESS Example This is usually called a sum-of-products (SOP) configuration ABcF C.S. Choy36 PRODUCT-OF-SUM (POS) CONFIGURATION ABcF C.S. Choy37 DESIGN ALTERNATIVE USING BOOLEAN ALGBRA Fully NAND Implementation F = B + A(C + D) C.S. Choy38 DESIGN ALTERNATIVE USING BOOLEAN ALGEBRA Fully NOR Implementation F = B + A(C + D) = B + AC + AD = B + A + C + A + D F = B + A(C + D) = B A (C + D) = B (A + C + D) =AB + B C+D = A + B + B + C + D</p>