4.3.6

Karnaugh Maps

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Karnaugh Maps

Karnaugh maps are diagrams used to simplify boolean expressions.

Drawing Karnaugh maps

Drawing Karnaugh maps

  • Solving a two-variable problem using a Karnaugh map involves creating a grid as shown:
    • The ‘0’ and ‘1’ down the left of the grid refer to ‘not A’ and 'A'.
    • The the ‘0’ and ‘1’ along the top of the grid refer to ‘not B' and 'B'.
Example

Example

  • Simplify the expression:
    • A.B‾+A‾.B+A.BA.\overline{B}+\overline{A}.B+A.B
  • We start by splitting it into three sets of brackets using the associative rule:
    • (A.B‾)+(A‾.B)+(A.B)(A.\overline{B})+(\overline{A}.B)+(A.B)
Fill in the table

Fill in the table

  • The first of these brackets refers to ‘A AND NOT B’.
  • That means we put a ‘1’ in the box with the coordinates (A1, B0) as shown.
  • The second bracket refers to ‘NOT A AND B’ .
  • That means we put a '1' in the box with coordinates (A0, B1).
  • The third bracket means ‘A AND B’, so we put a one in the box with coordinates (A1, B1).
Group

Group

  • The final stage involves grouping the '1's into sets of either 2, 4 or 8.
  • In this example, there are only three '1's in total, but they can be grouped into two pairs as shown.
  • The green pair both have the same value for A (1) and the blue pair have the same value for B (1).
  • So our original expression simplifies to A+BA + B.

Karnaugh Maps

We can also construct Karnaugh maps to simplify expressions with 3 or 4 variables.

Three variables

Three variables

  • With a three-variable problem, the grid is expanded by adding variable C to the top.
  • This means each column represents a combination of the variables B and C.
    • In the second column of the map shown, labelled 01.
    • The 0 refers to B and the 1 refers to C.
  • The order of these combinations follows the rules of Gray code.
  • This means that each combination is only a single bit different from the combinations either side of it.
Four variables

Four variables

  • For four variable Karnaugh maps, we do the same as for three variable but the rows now come to represent another combination of variables.
  • Each row is labelled using the Gray code as before.
Simplification

Simplification

  • In this example, the ‘1’s have been grouped into a pair (green) and a set of four (blue).
  • This is interpreted as (A AND B AND NOT C) OR (C AND NOT D) with the expression:
    • (A.B.C‾)+(C.D‾)(A.B.\overline{C})+(C.\overline{D})
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Boolean Algebra

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Practice questions on Karnaugh Maps

Can you answer these? Test yourself with free interactive practice on Seneca — used by over 10 million students.

  1. 1
    Which expression is represented by the Karnaugh map shown?Multiple choice
  2. 2
    Which expression is represented by the Karnaugh map shown?Multiple choice
  3. 3
    Which expression is represented by the Karnaugh map shown?Multiple choice
  4. 4
    Which expression is represented by the Karnaugh map shown?Multiple choice
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Boolean Logic

Boolean Identities