By T.J. Stonham

1 Numerical illustration of information.- 2 Operations on binary data.- three Combinational common sense design.- four Sequential good judgment fundamentals.- five layout of sequential common sense circuits.- 6 The electronic system.- 7 useful electronic circuits.- solutions to difficulties.

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3. Show that A + A = 1 using logical reasoning. LetF = A+A The equation reads 'iff' reads 'if and only if'. '. F is 1 iff A is 1 or A is 0 But A is binary and must be either 0 or 1, so one of the conditions for A is always satisfied. :. Fis always 1 :. A + A = 1 In a Venn diagram, the binary variables of a function are represented as overlapping areas in a universe. The OR function is the combination or union of areas and the AND function is the interseetion or common part between two or more overlapping areas.

We shall also examine graphical and tabular methods of simplifying logic equations that replace the absorption and minimization theorems. De Morgan's theorems however are important in the design of logic systems. They do not have a parallel in arithmetic and must be leamt. Verification oj Boolean Theorems Boolean theorems can be verified by logical reasoning or by demonstrating equivalence relationships on a Venn diagram. 3. Show that A + A = 1 using logical reasoning. LetF = A+A The equation reads 'iff' reads 'if and only if'.

ANO, OR, and NOT functions can be performed with NOR gates only. The equivalent circuits are NOT OR ~F B ~F B 43 AND ~F F B Remember NANO form is derived from 1st canonical form. NOR form is derived from 2nd canonical form. The NOR form can be obtained from the minimal 2nd canonical form by replacing the OR/AND gates with NORs. If however a variable or its inverse only passes through the output AND gate, an additional inverter must be introduced on that input line before replacing the gates with NORs.