# Karnaugh Maps

It may be neccessary to read the section on Boolean Algebra in order to understand some of the notation used within this article.

A Karnaugh Map makes simplification of truth tables into functions much easier, but it is quite an involved concept to learn. You get all the minterms, and draw a rectangle around them using the fewest rectangles possible that include all the selected minterms. A rectangle must have 2^N minterms within it (so 1,2,4,8,16 etc.) The size of Karnaugh map depends on the number of variables dealt with.

In order to begin teaching the concepts behind Karnaugh maps, I will start with a 2x1 grid. This would not actually be used, but gives an example of how Karnaugh Maps work. I will use the example of an invertor. If a 1 is put in, a 0 is given out. If a 0 is put in, a 1 is given out. I will use the term xn to describe the letter x followed by a number. xn would normally be replaced by a 0, a 1 or a don't care term, represented by a - as is shown in examples 2.1 and 2.2 below. As you can see, everything below the the A is the value for when A = 1. Everything which doesn't have an A above is the value for when A = 0. This may not make much sense yet, but should start to when we deal with larger maps.

Input | Output |
---|---|

A | X |

0 | X0 |

1 | X1 |

A | |
---|---|

x0 | x1 |

For the invertor example, this would have the values

Input | Output |
---|---|

A | X |

0 | 1 |

1 | 0 |

A | |
---|---|

1 | 0 |

Below, I've shown how the output(X) of the truth table (normally 0s or 1s) are mapped onto a 4x4 karnaugh map.

Example 3.1 Truth Table for 4x4 Karnaugh Map

Inputs | Output | |||
---|---|---|---|---|

A | B | C | D | X |

0 | 0 | 0 | 0 | X0 |

0 | 0 | 0 | 1 | X1 |

0 | 0 | 1 | 0 | X2 |

0 | 0 | 1 | 1 | X3 |

0 | 1 | 0 | 0 | X4 |

0 | 1 | 0 | 1 | X5 |

0 | 1 | 1 | 0 | X6 |

0 | 1 | 1 | 1 | X7 |

1 | 0 | 0 | 0 | X8 |

1 | 0 | 0 | 1 | X9 |

1 | 0 | 1 | 0 | X10 |

1 | 0 | 1 | 1 | X11 |

1 | 1 | 0 | 0 | X12 |

1 | 1 | 0 | 1 | X13 |

1 | 1 | 1 | 0 | X14 |

1 | 1 | 1 | 1 | X15 |

A | |||||
---|---|---|---|---|---|

B | |||||

x0 | x1 | x3 | x2 | ||

C | x4 | x5 | x7 | x6 | |

D | x12 | x13 | x15 | x14 | |

x8 | x9 | x11 | x10 |

In use, each occurance of X and a number (Xn) would be replaced with a 0, 1 or - (don't care) term.

The letters at the top and sides show where each of the input variables (A,B,C and D) equal 1. For the Karnaugh map to be translated into a function, it is neccessary to group similar terms. An example of this is shown below. The function (ABCD)(ABCD)(ABCD)(ABCD) is mapped below. It can be seen that are equal to 1 are shaded. All the shaded cells fall into both C and D. Therefore, the simplified function is CD. Don't care terms can be assigned to either a 1 or a 0 depending on which leads to a more simplified solution.

Truth Table for 4x4 Karnaugh Map

Inputs | Output | |||
---|---|---|---|---|

A | B | C | D | X |

0 | 0 | 0 | 0 | 0 |

0 | 0 | 0 | 1 | 0 |

0 | 0 | 1 | 0 | 0 |

0 | 0 | 1 | 1 | 0 |

0 | 1 | 0 | 0 | 0 |

0 | 1 | 0 | 1 | 0 |

0 | 1 | 1 | 0 | 0 |

0 | 1 | 1 | 1 | 0 |

1 | 0 | 0 | 0 | 0 |

1 | 0 | 0 | 1 | 0 |

1 | 0 | 1 | 0 | 0 |

1 | 0 | 1 | 1 | 0 |

1 | 1 | 0 | 0 | 1 |

1 | 1 | 0 | 1 | 1 |

1 | 1 | 1 | 0 | 1 |

1 | 1 | 1 | 1 | 1 |

A | |||||
---|---|---|---|---|---|

B | |||||

0 | 0 | 0 | 0 | ||

C | 0 | 0 | 0 | 0 | |

D | 1 | 1 | 1 | 1 | |

0 | 0 | 0 | 0 |