## What is 4 Variables Karnaugh's Map?

**4 Variables Karnaugh's Map** often known as 4 variables K-Map. It's an alternate method to solve or minimize the Boolean expressions based on AND, OR & NOT gates logical expressions or truth tables. The four variables A, B, C & D are the *binary numbers* which are used to address the min-term SOP of the Boolean expressions. The *gray code conversion method* is used to address the cells of KMAP table.

The min-term SOP is often denoted by either ABCD, 1s & 0s or *decimal numbers*. For example, the Boolean expression y = ∑{2, 6, 9, 11, 15} represents the place values of the respective cells which has the higher values (binary 1s). The y = ∑{2, 6, 9, 11, 15} can also be represented by y = ∑{0010, 0110, 1001, 1011, 1111} or y = ∑{ABCD, ABCD, ABCD, ABCD, ABCD}

A is the most significant bit (MSB) and B is the least significant bit (LSB). Each variable A, B, C & D equals to value 1. Similarly, each inverted variable A, B, C & D equals to 0. Any 4 *combinations* of A, B, C, D, A, B, C & D represents the place values of 0 to 15 to address the cells of table in KMAP solver.

For example, the combinations ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD & ABCD represents the binary values of 0000, 0001, 0010, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110 & 1111 respectively. The numeric or decimal equivalent for the combinations A, B, C, D, A, B, C & D represents the cell or place values from 0 to 15 in the table of K-Map solver. For example, the combinations ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD & ABCD represents 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 & 15 respectively.

Users may refer the below details to learn more about 4 variables Karnaugh's map or use this online calculator to solve the SOP or generate the complete work for minimum SOP for 4 variables A, B, C & D.

## How to Solve 4 Variables KMAP?

Users may refer the below rules & step by step procedure to learn how to find the minimum sum of products (SOP) for the Boolean expression using 4 variables A, B, C & D. When you try yourself solving the min-term SOP of for 3 variables, Users can use this online Karnaugh's map solver for 4 variables to verify the results of manual calculations.

step 1 Addressing the cells of KMap table

When using KMAP solver, generally users should be careful while placing the min-terms. Because, the addressing of min-terms in KMAP table is bit different. The order of the cells are based on the Gray-code method. Refer the below table & information gives the idea of how to group the KMAP cells together. For four variables, the location of the the cells of KMAP table as follows

__In Binary Form__

Row 1: 0000, 0001, 0011, 0010

Row 2: 0100, 0101, 0111, 0110

Row 3: 1100, 1101, 1111, 1110

Row 4: 1000, 1001, 1011, 1010

0000 | 0001 | 0011 | 0010 |

0100 | 0101 | 0111 | 0110 |

1100 | 1101 | 1111 | 1110 |

1000 | 1001 | 1011 | 1010 |

__In Decimal Form__

Row 1: 0, 1, 3, 2

Row 2: 4, 5, 7, 6

Row 3: 12, 13, 15, 14

Row 4: 8, 9, 11, 10

__In Variable (A, B, C & D) Form__

Row 1: ABCD, ABCD, ABCD, ABCD

Row 2: ABCD, ABCD, ABCD, ABCD

Row 3: ABCD, ABABCD, ABCD, ABCD

Row 4: ABCD, ABCD, ABCD, ABCD

step 2 Write the Boolean expression in the SOP form. Place 1s for those positions in the Boolean expressions and 0s for everything else.

step 3 Group the 1s. The counting of 1s in the group should be in the form of 2

^{3}, 2

^{4}, 2

^{2}and 2

^{1}. Therefore you can't group single 1s, three 1s, five 1s, six 1s, seven 1s, nine 1s, ten 1s, eleven 1s, twelve 1s, thirteen 1s, fourteen 1s & fifteen 1s. The possible combinations of grouping are sixteen 1s, eight 1s, four 1s and two 1s together.

step 4 When grouping of 1s the first and last columns are considered adjacent to each other. Similarly, the first & last rows are considered adjacent to each other. The four corner cells of the KMAP table also considered as adjacent to each other.

step 5 Check for sixteen 1s group and encircle the combination, if any.

step 6 Check for eight 1s group and encircle the combination, if any.

step 7 Check for four 1s group and encircle the combination, if any.

step 8 Check for two 1s group and encircle the combination, if any.

step 9 Find the appropriate product term for each combinations.

step 10 Add all the product terms brings the Minimum SOP of the given Boolean expression