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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
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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.
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NOT via NAND
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AND via NAND
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OR via NAND
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Summary of Boolean algebra
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Basic theorems
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Boolean expressions
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Absorption
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Sum-of-products form absorbed
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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.
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Example absorbed
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Complete sum-of-products forms
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Completing sum-of-products forms
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Example
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Minimal sum-of-products
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Prime implicants
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Theorem
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Karnaugh map (Geometric method)
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Case of two variables
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Prime implicants
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Ex 1
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Ex 2
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Ex 3
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Case of three variables
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Largest implicants
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Ex 1
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Ex 2
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Ex 3
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Problem 1
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Problem 2
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Problem 3
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Problem 4
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