Digital Computer Design. Logic, Circuitry, and Synthesis by Edward L. Braun

By Edward L. Braun

Electronic computing device layout, common sense, Circuitry, and Synthesis

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Sample text

4. THE REPRESENTATION OF SWITCHING FUNCTIONS 49 Operators only. An analogous statement is that any switching function can be constructed utilizing these three switching functions. 6, represents the NOT operator. Actually, AND, OR, and NOT do not represent a minimal set of independent operators. This can be proved by showing that the OR function can be gen­ erated by means of AND and NOT functions, and also that the AND function can be generated by means of OR and NOT functions. The proof of the first case is simply that A + Β = (ÄB).

3-14) by the principle of dualization. It states that any Boolean equation can be written as a logical product of logical sums* (generally abbreviated, for convenience, to "a product of sums") each of which contains all input variables. A representation of this type is said to be in conjunctive normal form. , (3-15) Some simple identities that can be obtained by means of the duaUty theorem are {AB) = ΑΛ-Ε {A+B)^ (3-16) AB (3-17) ( J ) = A, (3-18) Some tautologiest useful for simplifying elemental forms are listed below: A+ A = \ (3-19) ÄA = Q (3-20) A= A+A= Α+Α-{Ά A = AA = AAA = .

A/B 0 1 0 0 1 0 Negation A/B 0 0 1 NOR F2 0 1 0 0 0 0 0 1 = 0 1 ANDF9 = AB A/B 0 1 1 0 0 1 0 0 1 0 1 = 0 AB A/B 0 1 0 0 1 0 Comparison Fio = AB -\- AB 0 1 1 0 1 0 0 1 Inhibiting gate F3 = AB A/B A/B 0 1 1 0 A/B Single identity F n 0 1 1 1 1 0 0 1 Single negation F4 = A A/B Conditional generator Fn = (Aß) 48 3. A/B BOOLEAN ALGEBRA A/B 0 1 0 0 0 0 1 1 0 1 0 1 Single identity F^g = A Inhibiting gate F5 = AB A/B 0 1 A/B 0 1 0 1 0 0 1 0 1 1 0 1 1 1 Conditional generator Fu = (AB) Single negation Fe = Β A/B A/B 0 1 1 0 1 0 1 1 0 1 0 1 Exclusive or Fj = Aß -\- AB orFis = A Β A/B 0 1 A/B 0 1 0 1 1 0 1 1 1 1 0 1 1 1 Tautology Fie = 1 Sheffer stroke Fg = AB Fi Note that the functions £Ί and E4 produce outputs independent of the input, and may be thought of as a 0 generator and a 1 generator, respec­ tively.

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