Chapter 5 Boolean Algebra and Reduction Techniques 1

5-5 DeMorgan’s Theorem Used to simplify circuits containing NAND and NOR gates A B = A + B A + B = A B 2

DeMorgan’s Theorem Break the bar over the variables and change the sign between them –Inversion bubbles - used to show inversion. Use parentheses to maintain proper groupings Results in Sum-of-Products (SOP) form 3

Figure 5.38 De Morgan’s theorem applied to NAND gate produces two identical truth tables. 4

Figure 5.39 (a) De Morgan’s theorem applied to NOR gate produces two identical truth tables; 5

More examples 6

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Bubble Pushing 1. Change the logic gate (AND to OR or OR to AND) 2.Add bubbles to the inputs and outputs where there were none and remove original bubbles 19

5-7 The Universal Capability of NAND and NOR Gates An inverter can be formed from a NAND simply by connecting both NAND inputs as shown in Figure

More examples Figure 5-69 Forming an AND with two NANDs 21

Figure 5-70, 5-71 (Equivalent logic circuit using only NANDs 22

Fig 5-72 External connections to form the circuit of Fig

Figure 5-74 Forming a NOR with four NANDs Figure 5-73 Forming an OR from there NANDs. 24

Discussion Point The technique used to form all gates from NANDs can also be used with NOR gates. Here is an inverter: Form an inverter from a NOR gate

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5-8 AND-OR-INVERT Gates for Implementing Sum-of-Products Expressions 30

AND-OR-INVERT Gates for Implementing Sum-of-Products Expressions 30 31

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