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NAND as a complete system and Karnaugh Maps

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Presentation on theme: "NAND as a complete system and Karnaugh Maps"— Presentation transcript:

1 NAND as a complete system and Karnaugh Maps
Discrete Systems I Lecture 11 NAND as a complete system and Karnaugh Maps Profs. Koike and Yukita

2 NAND as a complete system
Using AND, OR, and NOT, we can construct any Boolean functions out of them. We will show that the NAND gate constitutes a complete system by itself. This means that if you once got an efficient implementation of the NAND gate you can construct the whole Boolean algebra.

3 NOT via NAND

4 AND via NAND

5 OR via NAND

6 Summary of Boolean algebra

7 Basic theorems

8 Boolean expressions

9 Absorption

10 Sum-of-products form absorbed

11 Finding sum-of-products form
Algorithm Input: A Boolean expression E. Output: A sum-of-products expression equivalent to E. Step 1: Convert E to an expression in which complement operations are only on literals. Step 2: Distribute so that E will be a sum of products. Step 3: Transform each product in E to a fundamental product. Step 4: Absorb any products as far as possible.

12 Example absorbed

13 Complete sum-of-products forms

14 Completing sum-of-products forms

15 Example

16 Minimal sum-of-products

17 Prime implicants

18 Theorem

19 Karnaugh map (Geometric method)

20 Case of two variables

21 Prime implicants

22 Ex 1

23 Ex 2

24 Ex 3

25 Case of three variables

26 Largest implicants

27 Ex 1

28 Ex 2

29 Ex 3

30 Problem 1

31 Problem 2

32 Problem 3

33 Problem 4


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