Title: Simplifying Boolean Expressions Using KMap Method
1 Simplifying Boolean Expressions Using K-Map Method
CS 3402-- Digital Logic Design
2 Simplifying Switching Functions
SOP and POS expressions
gt 2-level circuits
Minimum SOP/POS expression
Minimize the number of literals
Minimum number of terms
Algebraically I.e. using the axioms and theorems of Boolean algebra.
3 Simplifying Algebraically
Simplify F(XYZ) X YZ XYZ XYZ
Simplify F(ABC) ABC ABC ABC
4 Simplifying Switching Functions K-Map
XY XY Y
Definition Logical Adjacency
Two terms are logically adjacent iff they differ in only one literal the literal is complemented in one term and non-complemented in the other.
Two Logically adjacent terms can be combined into one term consisting of only the common literals
5 Simplifying Boolean Expressions
Example XY XY
6 Karnaugh-Map (K-Map)
2-dimensional representation of a truth table.
Logically adjacent terms are physically adjacent in the map.
2-Variable Functions F(XY) XY XY
X Y F(XY) 0 0 0 1 1 0 1 1 m0 m1 m2 m3 7 2-Variable K-Map
XY XY XY XY
Logically adjacent cells are physically adjacent in the k-map
Each cells has two adjacent cells
m0 m2 m1 m3 8 Function Minimization Using K-Maps
1. Each square (minterm) in a k-map of 2 variables has 2 logically adjacent squares each square in a 3-variable k-map has 3 adjacent squares etc.
2. Combine only the minterms for which the function is 1.
3. When combining terms on a k-map group adjacent squares in groups of powers of 2 (I.e. 2 4 8 etc.). Grouping two squares eliminates one variables grouping 4 squares eliminates 2 variables etc.
Cant combine a group of 3 minterms
9 Function Minimization Using K-Maps
4. Group as many squares together as possible the larger the group is the fewer the number of literals in the resulting product term
5. Select as few groups as possible to cover all the minterms of the functions. A minterm is covered if it is included in at least one group. Each minterm may be covered as many times as it is needed however it must be covered at least once.
6. In combining squares on the map always begin with those squares for which there are the fewest number of adjacent squares (the loneliest squares on the map).
Implicant a product term that could be used to cover one or more minterms
Prime Implicant A product term obtained by combining the maximum number of adjacent squares in the map.
Essential Prime Implicant A prime implicant that covers at least one minterm that is not covered by any other prime implicant.
All essential prime implicants must be included in the final minimal expression.
11 Definitions (Cont.)
Cover of function is a set of prime implicants for which each minterm of the function is covered by at least one prime implicant.
All essential prime implicants must be included in the cover of a function.
12 Algorithm for Deriving the Minimal SOP
1. Circle all prime implicants on the k-map
2. Identify and select all essential prime implicants
3. Select a minimum subset of the remaining prime implicants to cover those minterms not covered by the essential prime implicants.
13 Four Variable K-map 14 Five Variable K-map 15 More Examples
Find min. SOP and POS expression for each of the following functions
F3(ABCDE) m(012456131516182224 2629)
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