1 / 9

Multiple representations

a. z. b. 0 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 0. a. z. 0 0 1 0 0 1 0 0 1 0 1 1 1 1 0 0. b. Multiple representations. Consider this logic diagram…. Here’s another logic diagram….

rhian
Download Presentation

Multiple representations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. a z b 0 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 0 a z 0 0 1 0 0 1 0 0 1 0 1 1 1 1 0 0 b Multiple representations Consider this logic diagram… Here’s another logic diagram… The truth table is the same – they implement the same logical function!

  2. Logic Minimization • Logic minimization is the process of converting a circuit to an equivalent circuit with the smallest number of gates or inputs • Reduce the number of gates • Reduce the number of inputs • Reduce the number of levels between inputs and outputs • Why bother? The circuits are equivalent • Fewer gates - cheaper • Fewer inputs - faster • Too many inputs may not work in some technologies • Fewer levels - faster circuits

  3. A B C Z 0 0 0 0 0 0 1 1 Z 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 A B 1 1 0 1 C 1 1 1 0 Z 2 C Z A 3 B C Logic Functions: Alternative Realizations A B C A C Two-Level Realization (inverters don't count) B C Multi-Level RealizationAdvantage: 2-input Gates instead of 3-input Complex Gate: XOR Advantage: Fewest Gates

  4. Boolean* Axioms AND OR Duality: Replace 1’s with 0’s and OR with AND to get another valid axiom Note: These are simply truth-table definitions of AND, OR, and NOT Inversion * George Boole, An Investigation of the rules of thought, 1854.

  5. Single-Variable Boolean Theorems AND OR Theorems can be derived from axioms by proofs Perfect induction proof: Enumerate all possibilities Prove x + 1 = 1 1. Assume x=0 Ö Axiom: 0+1=1 Inversion 2. Assume x=1 Ö Axiom: 1+1=1 Cases 1 and 2 cover all possibilities for x Ö

  6. a b a+b (a+b)’ a’ b’ a’b’ 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 1 0 0 0 0 a b ab (ab)’ a’ b’ a’+b’ 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 DeMorgan’s Law NOR is equivalent to AND with inputs complemented NAND is equivalent to OR with inputs complemented

  7. A A A A B B B B A A A A B B B B A A A A B B B B DeMorgan’s Law, in Picture Form

  8. + + + Using DeMorgan’s Law Simplifying an expression using DeMorgan’s law means to reduce the scope of inversions (bars) to only a single variable 1. Look for inversions and find their scope + • 2. Apply DeMorgan’s Law • a. Remove inversionb. Swap AND/OR in scopec. Invert terms in scope

  9. Multi-Variable Boolean Theorems Commutivity Associativity Distributivity Absorption Combining Adsorption Shaded expressions are true for general mathematical expressions

More Related