### lec6

```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
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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
```