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

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.

NOT via NAND

AND via NAND

OR via NAND

Summary of Boolean algebra

Basic theorems

Boolean expressions

Absorption

Sum-of-products form absorbed

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.

Example absorbed

Complete sum-of-products forms

Completing sum-of-products forms

Example

Minimal sum-of-products

Prime implicants

Theorem

Karnaugh map (Geometric method)

Case of two variables

Prime implicants

Ex 1

Ex 2

Ex 3

Case of three variables

Largest implicants

Ex 1

Ex 2

Ex 3

Problem 1

Problem 2

Problem 3

Problem 4