1. Chapter 5 Boolean Algebra and Reduction Techniques 1

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

3. 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

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

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

6. More examples 6

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19. 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

20. 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 5-68. 20

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

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

23. Fig 5-72 External connections to form the circuit of Fig 5-71. 23

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

25. 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. 29 25

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

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

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