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Simplify the Boolean expression A A B.
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Solution First make use of item (17), then item (9); thus " "" " " " A A B A 1 B A; answer
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Binary Arithmetic
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Note: In regard to the use of item (9) above, it should be understood that the expression 1 A 1, is true regardless of what Boolean expression A might actually represent; for example, 1 ABCD 1, or, 1 B C D 1, and so on. We thus regard the expression 1 A 1 as the generic relationship, valid for whatever the actual form of A might be. The same statement holds for the other relationships in the table; for example, the Boolean relationship AB C D has the basic form of item (18); and thus, by item (18), we have that AB C D AB C D Example 9
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Simplify the Boolean expression A B C AB.
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Solution Using item (18), write the given expression in the form AB C AB AB AB C the or operation is commutative to which now apply the generic form of item (16) to get AB C, answer. It should be noted that the new expression, AB C, does exactly the same switching job as the original expression but requires only one and and one or operation, whereas the original expression requires two and operations, two or operations, and two not (inverter) operations. Example 10
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Given Z ABC D A B C , nd a simpler expression for Z .
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Solution " " " " First, by item (18), A B C AB C ABC, so that the given problem now becomes Z ABC ABC D which is the generic form of item (16); and thus, Z ABC D, answer, which represents a considerably simpler switching network than the original expression. Problem 273 A A A A B B B Problem 274 AAABBCCCC Problem 275 A B C ABC 1 Problem 276 Using the table of basic theorems, show that """ A B C ABC
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CHAPTER 12 Binary Arithmetic
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Problem 277 Using the table of basic theorems, show that " " " A B C ABC Note: In problems 278 through 284, which follow, simplify the given Boolean equations. Try to minimize both the number of switching operations and the number of not operations needed. Problem 278 Z A AB Problem 279 " Z A AB AB C Problem 280 Z A A B B C Problem 281 " "" " Z AC BC B C A C B C Problem 282 " " Z B C D B D Problem 283 " """ Z ABC A BC A B C Problem 284 """ " Z ABCD A C D Problem 285 " Write A A B in a form without not operations. Problem 286 "" " "" Write the equation Z A B C A B C A B C in a form that will require only one not operation.
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Digital Logic Symbols and Networks
A digital LOGIC NETWORK is an arrangement of and, or, and not devices that will always produce a required form of output signal for a given set of input binary signals. For example, let A; B, and C denote the 0, 1 values of three binary signals, and " suppose it is necessary to generate the output signal, Z AB C . The logic network needed to perform this particular operation is shown in block diagram form in Fig. 317. We ll not concern ourselves presently with the physical details of what might be actually inside each of the above boxes.
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Fig. 317
In Fig. 317 we ve designated the function of each block with the words AND, OR, and NOT. Instead of words, however, certain symbolic forms have been universally adopted to indicate these operations, as follows.
The above symbols should be committed to memory. In regard to the AND and OR symbols, it should be understood that more than just two input lines can be used; for instance, if, say, three binary signals, A; B, and C, are involved, this would be indicated by using three input lines, thus
The triangular symbol by itself,
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