By A. Potton (auth.)
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Extra resources for An Introduction to Digital Logic
15 represents a function with two terms which are (A· /I) and (A· D). In cases of loops of eight cells in a sixteen cell map, four cells in an eight cell map or two cells in a four cell map, the loop is identified by a single variable. If such a loop is identified by a single variable X, in the procedure described here, instead of two or more variables linked by a NAND operation, the single variable is inverted to form the term. A loop which covers the entire X area of a map therefore provides a term X whereas if the loop is completely outside the X area, it contributes a term X.
The next section demonstrates how application of these rules can greatly assist the design of economical systems to perform simple logical operations. Problems 1 Without using truth tables prove that A · (B +if) =A. 2 Use the results of problem 1 to prove the following relationship A·B+A·B+C·B+C·B=A+C 3 Without using truth tables, prove that X·Y·Z+X·Y·Z+X·Y·Z=X·Y+X·Z (Hint: since A +A =A then X· Y· Z=X· Y· Z+X· Y· Z) 4 Use truth tables to prove both forms of the De Morgan relationship. 5 Show by means of truth tables that De Morgan's relationships can be extended to three variables.
The systems given in this section are not necessarily the most economical and it is quite possible that the number of gates used could be reduced. The problem of designing the most economical or 'minimal' systems is however considerably more complicated than merely designing a simple system which performs the required operation. This is discussed further in the next chapter. [~] F [e. 5. 5 inputs c output F A 8 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 2 A control panel has three lamps A, B, C and two switches X and Y.