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Kmaps Digital Logics, Summaries of Mathematics

Mathematics

KmapsDigitalLogicDesign Sum of products

Typology: Summaries

2025/2026

Uploaded on 03/08/2026

mohammed-halai
mohammed-halai 🇺🇸

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Karnaugh Maps (K-Maps)
- K-Maps are graphical tools used to obtain simplified expressions for logic functions.
- For n variables, there will be 2n cells (squares) in the K-Map. (For example, for n=2
there are 4 cells, for n=4, there are 16 cells, …).
Examples of K-Maps:
n=2 (X, Y) n=3 (X, Y, Z) n=4 (X, Y, Z, W)
n=5 (X, Y, Z, W, V)
X=0 X=1
n=6 (X, Y, Z, W, V, U)
XY=00 XY=01
XY=10 XY=11
pf3
pf4
pf5

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Karnaugh Maps (K-Maps)

  • K-Maps are graphical tools used to obtain simplified expressions for logic functions.
  • For n variables, there will be 2 n^ cells (squares) in the K-Map. (For example, for n= there are 4 cells, for n=4, there are 16 cells, …). Examples of K-Maps: n=2 (X, Y) n= 3 (X, Y, Z) n= 4 (X, Y, Z, W) n= 5 (X, Y, Z, W, V) X=0 X= n= 6 (X, Y, Z, W, V, U) XY=0 0 XY=0 1 XY=1 0 XY=1 1
  • Each cell in a K-Map represents one row of the truth table. Alternatively we can say each cell represents one minterm (or maxterm) of a single output function.
  • Each pair of adjacent cells corresponds to a pair of input combinations (minterms /maxterms) that differ in only one variable which is uncomplemented in one cell and complemented in the other.
  • As we learned, any Boolean function can be expressed as a sum of minterms. Therefore, a Boolean function is expressed graphically in the K-map by the squares for which the function has value 1. Saying it differently, one basically places “ 1 ” in the cells corresponding to the minterms that are included in the function. Example: F (X,Y,Z)= ∑ (^) *,+,,( 0 , 1 , 4 , 7 )= X'.Y'.Z' + X'.Y'.Z + X.Y'.Z' + X.Y.Z can be expressed in a K-Map as : or
  • Similarly, any Boolean function can be expressed as a product of maxterms. If this is of interest, then a Boolean function can be expressed graphically in the K-map by the squares for which the function has value 0. Saying it differently, one basically places “ 0 ” in the cells corresponding to the maxterms that are included in the functions. Example: F (X,Y,Z)= ∏ (^) *,+,, ( 2 , 3 , 5 , 6 )= (X+Y'+Z) (X+Y'+Z') (X'+Y+Z') (X'+Y'+Z) can be expressed in a K-Map as : sdsdor

Finding Minimal Sum-of-Product (SOP) Using K-Maps

Goal: to reduce the number and size of gates that are needed to build the function, which will result in chips that are faster and more efficient while consuming less power (longer battery life). Process:

  1. Based on the number of variables in the function, draw the corresponding K-Map.
  2. Place “1” in cells corresponding to the minterms of the function.
  3. Start by identifying the largest possible sets of adjacent “1-cells”, and identify all prime implicants. Note: number of circled cells must be power of 2 (i.e. 1, 2, 4, 8, …). You can cross edges across the borders of the map.
  4. Identify all the essential prime implicants.
  5. Identify prime implicants that are needed to include minterms in the function that are not covered by the identified essential prime implicants.
  6. The minimal sop is obtained by summing products found in steps 4 and 5. Example: Find minimal SOP for the function F (A,B,C,D)= ∑^2 , 3 , 4 , 5 ( 0 , 5 , 10 , 11 , 12 , 13 , 15 ).

Example: Find the Boolean expression that identifies prime numbers between 0 and 15. Example: Find the Boolean expression that identifies prime numbers between 0 and 31.