Karnaugh map (k-map) explanation

K-maps, often referred to as Karnaugh maps, are a type of visual used in digital logic to optimize logic circuits and streamline Boolean expressions. They offer a methodical approach for classifying minterms (or maxterms) according to their binary patterns. You can discover necessary prime implicants and obtain a minimal sum-of-products (SOP) or product-of-sums (POS) expression by grouping neighboring cells together with 1s. Karnaugh maps are useful in lowering the complexity of digital circuit design and the number of logic gates. 

Let's go over a straightforward Karnaugh map example with a 2-variable Boolean statement.

Suppose we have the following truth table for a Boolean function F(A, B):

| A | B | F(A, B) |

|---|---|--------|

| 0 | 0 |   0    |

| 0 | 1 |   1    |

| 1 | 0 |   1    |

| 1 | 1 |   0    |

The variables A and B are used as the row and column headings, respectively, in a 2x2 grid to organize the truth table values in order to create the Karnaugh map:

|   | 0 | 1 |

|---|---|---|

| 0 | 0 | 1 |

| 1 | 1 | 0 |

We now combine neighboring cells that contain 1s. In order to cover as many 1s as possible, we wish to make groups of 1s whose size is a power of 2 (1, 2, 4, 8, etc.). The two 1s in this instance can be combined:

|   | 0 | 1 |

|---|---|---|

| 0 |   | 1 |

| 1 | 1 |   |

Since there are just two 1s, we can group those two cells together. The Boolean expression uses this group to denote a minterm. By converting the group back to its Boolean form, we can now determine the abbreviated Boolean expression.

The group is given by the minimal expression F(A, B) = A' B.

Therefore, F(A, B) = A' B is the abbreviated Boolean expression for the provided truth table.

Let's go over yet another illustration of a Karnaugh map with a Boolean statement with three variables.

Let's say we have the truth table below for the Boolean function G(A, B, C):

| A | B | C | G(A, B, C) |

|---|---|---|------------|

| 0 | 0 | 0 |     0      |

| 0 | 0 | 1 |     1      |

| 0 | 1 | 0 |     1      |

| 0 | 1 | 1 |     0      |

| 1 | 0 | 0 |     1      |

| 1 | 0 | 1 |     0      |

| 1 | 1 | 0 |     0      |

| 1 | 1 | 1 |     1      |

We arrange the truth table values in a 2 x 4 grid, with variables A and B serving as row headings and variable C serving as a column heading, to create the Karnaugh map:

|   | 00 | 01 | 11 | 10 |

|---|----|----|----|----|

| 0 |  0 |  1 |  0 |  1 |

| 1 |  1 |  0 |  1 |  0 |

We now combine neighboring cells that contain 1s. Once more, we aim to cover as many 1s as we can by forming groups of 1s whose size is a power of 2.

|   | 00 | 01 | 11 | 10 |

|---|----|----|----|----|

| 0 |  0 |  1 |  0 |  1 |

| 1 |  1 |  0 |  1 |  0 |                                        

The minterm for this group is G(A, B, C) = A'B'C + AB'C + ABC + A'BC.

For the truth table provided, the shortened Boolean equation is G(A, B, C) = A'B'C+ AB'C' + ABC + A'BC'.