Using Boolean algebra to simplify Boolean expressions can be difficult and may lead to solutions which, though they appear minimal, are not. The Karnaugh map provides a simple and straight-forward method of minimising boolean expressions which represent combinational logic circuits. A Karnaugh map is a pictorial method of grouping together expressions with common factors and then eliminating unwanted variables.
A Karnaugh map is a two-dimensional truth-table. Note that the squares are numbered so that the binary representations for the numbers of two adjacent squares differ in exactly one position.
Rules for Grouping together adjacent cells containing 1's
- Groups must contain 1, 2, 4, 8, 16 (2n) cells.
- Groups must contain only 1 (and X if don't care is allowed).
- Groups may be horizontal or vertical, but not diagonal.
- Groups should be as large as possible.
- Each cell containing a 1 must be in at least one group.
- Groups may overlap.
- Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.
- There should be as few groups as possible.
Obtaining Product Terms
- If A is a variable that has value 0 in all of the squares in the grouping, then the complemented form A is in the product term.
- If A is a variable that has value 1 in all of the squares in the grouping, then the true form A is in the product term.
- If A is a variable that has value 0 for some squares in the grouping and value 1 for others, then it is not in the product term.
Karnaugh Map Calculator
- Click on the buttons in the Truth Table or in the Karnaugh Map to change the value.
- Mouse over minterm components of the function F to see how the minterm is formed in the Karnaugh Map.
Application of Karnaugh Map
Karnaugh Maps are used to solve the following type of problem.
Bank Alarm System
A bank wants to install an alarm system with 3 movement sensors.
To prevent false alarms produced by a single sensor activation, the alarm will be triggered only when at least two sensors activate simultaneously.
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