c.s. choy21 boolean algegra the mathematics of logic boolean variables have only two possible values...

Download 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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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