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Chapter 5 Boolean Algebra and Reduction Techniques 1 5-5 DeMorgan’s Theorem • Used to simplify circuits containing NAND and NOR gates • AB=A+B • A+B=AB 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 7 8 9 10 11 12 13 14 15 16 17 18 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 5-68. 20 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 5-71. 23 Figure 5-73 Forming an OR from there NANDs. Figure 5-74 Forming a NOR with four 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. 25 29 26 27 28 29 5-8 AND-OR-INVERT Gates for Implementing Sum-of-Products Expressions 30 AND-OR-INVERT Gates for Implementing Sum-of-Products Expressions 31 30 32