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the small circle at the output side of the triangle that designates that the ampli er is used in the NOT or inverting mode. If the output of an AND network is passed through a NOT network, the result is NOT AND (abbreviated NAND), illustrated below to the left.
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The two-symbol NAND drawing (left above), is often expressed in a shortened form by the simple addition of a small not circle at the output side of the AND symbol, as shown to the right above. In the same way, if the output of an OR network is passed through a NOT network, the result is NOT OR (abbreviated NOR), illustrated to the left below, with the simpli ed version shown in the gure on the right below.
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CHAPTER 12 Binary Arithmetic
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Now consider the following. Suppose we are given an array of input signals, in the form of on-or-o pulses representing the binary digits 1 and 0, and suppose we must nd a switching network that will produce a desired result. In other words, the problem is, given a truth table, FIND A DIGITAL SWITCHING NETWORK that will satisfy the given truth table. One procedure that can be used to nd such required circuitry is to begin by writing down the basic or elemental Boolean equation for the given truth table. The elemental equation for a given truth table is a Boolean AND-OR relationship in which each AND term contains all the variables. This means that, if, for example, A denotes one of the " variables, then either A or not A (A or A) must appear in each of the and terms of the equation. For instance, if we are dealing with, say, three binary input signals, denoted by A; B, and C, then the elemental Boolean equation for a required switching system will be of the AND-OR form. " """ " "" " Z ABC A B C AB C A B C A BC A B C and likewise for any number of input variables, A; B; C; D; . . ., in which only those AND terms that will produce an output signal will be used; that is, only those AND terms for which Z 1 will be used. Consider the following two examples. Example 11
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Suppose three binary signals, denoted by A; B, and C, are to be switched in such a way as to satisfy the truth table A 1 1 1 1 B 1 1 0 0 C 1 0 1 0 Z 0 0 1 0 A 0 0 0 0 B 1 1 0 0 C 1 0 1 0 Z 1 0 1 0
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Write the elemental equation for the required switching network and simplify as much as possible.
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Solution In accordance with the foregoing rule, inspection of the given truth table shows that the elemental equation is " "" " Z A B C A BC A B C Now, while the above elemental form will do the required switching, it has the disadvantage of requiring two NOT circuits, three AND circuits, and one OR circuit. An equivalent but simpler circuit can, however, be found by applying the Boolean theorems to the above elemental expression; let us begin by factoring out the C signal; thus "" " " Z A B A B A B C hence " " " " Z A B A C A A B C because " "" " " " " A B A B A B B A 1 A; from the theorems:
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" A AB A B therefore " " " " A AB A B
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The last expression for Z can now be simpli ed further, as follows. By item (16),
" " " " " because A A B has the same basic form as A A B, with A written in place of A and B written in place of B. Therefore the last expression for Z becomes " " Z A B C AB C; by item 18 ; final answer which, using graphic block diagrams, is drawn as in Fig. 318.
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