Table of Contents
Combinational logic
Possible logic functions of two variables
Cost of different logic functions
Minimal set of functions
An algebraic structure
Boolean algebra
Logic functions and Boolean algebra
Axioms and theorems of Boolean algebra
Axioms and theorems of Boolean algebra (cont’d)
Axioms and theorems of Boolean algebra (cont’)
Axioms and theorems of Boolean algebra (cont’)
Proving theorems (rewriting)
Proving theorems (perfect induction)
A simple example
Apply the theorems to simplify expressions
From Boolean expressions to logic gates
From Boolean expressions to logic gates (cont’d)
From Boolean expressions to logic gates (cont’d)
Waveform view of logic functions
Choosing different realizations of a function
Which realization is best?
Which is the best realization? (cont’d)
Are all realizations equivalent?
Implementing Boolean functions
Canonical forms
Sum-of-products canonical forms
Sum-of-products canonical form (cont’d)
Product-of-sums canonical form
Product-of-sums canonical form (cont’d)
S-o-P, P-o-S, and de Morgan’s theorem
Four alternative two-level implementations of F = AB + C
Waveforms for the four alternatives
Mapping between canonical forms
Incompleteley specified functions
Notation for incompletely specified functions
Simplification of two-level combinational logic
The uniting theorem
Boolean cubes
Mapping truth tables onto Boolean cubes
Three variable example
Higher dimensional cubes
m-dimensional cubes in a n-dimensional Boolean space
Karnaugh maps
Karnaugh maps (cont’d)
Adjacencies in Karnaugh maps
Karnaugh map examples
More Karnaugh map examples
Karnaugh map: 4-variable example
Karnaugh maps: don’t cares
Karnaugh maps: don’t cares (cont’d)
Design example: two-bit comparator
Design example: two-bit comparator (cont’d)
Design example: two-bit comparator (cont’d)
Design example: 2x2-bit multiplier
Design example: 2x2-bit multiplier (cont’d)
Design example: BCD increment by 1
Design example: BCD increment by 1 (cont’d)
Definition of terms for two-level simplification
Examples to illustrate terms
Algorithm for two-level simplification
Algorithm for two-level simplification (example)
Combinational logic summary
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