- K Map Definition: A Karnaugh map (K-map) is defined as a visual tool used to simplify Boolean expressions by organizing values into a grid.
- Populate the K-map: Fill the K-map cells with ones for each product term and zeros for others to set up the simplification process.
- Form Groups: Group consecutive ones to simplify the expression, following rules for grouping size and shape.
- Boolean Expression: Extract simplified Boolean expressions for each group, combining them into a Sum-of-Products (SOP) form.
- Simplify Boolean Expression Using K-Map: This method involves organizing and grouping terms in a K-map to derive a more straightforward Boolean expression.
Minterm Solution of K Map
A K-map is a grid that simplifies a Boolean function by grouping adjacent 1s (minterms) or 0s (maxterms).
Step 1: Initiate
Convert the given Boolean expression to canonical form.
Step 2: Populate the K-map
Write 1 in each cell that belongs to a product term. Write 0 in the rest.
Step 3: Form Groups
- Group consecutive 1s in the K-map cells (green boxes).
- Each group should hold as many 1s as possible and no empty cell.
- The number of 1s in a group must be a power of 2, so a group can contain
- Form groups from largest to smallest: try 8 (octet) first, then 4 (quad), then 2, then isolated 1s.
- Groups must be horizontal, vertical, square or rectangular. Diagonal groups of 1s are not allowed.
- The same cell may sit in more than one group only when that makes a group larger.
- Cells on opposite edges of the map are adjacent and may be grouped together.
- Don’t care conditions are to be considered only if they aid in increasing the group-size (else neglected).
Step 4: Obtain Boolean Expression for Each Group
Write each group from the input variables that stay constant across its cells. In the figure below, Group 1 has two 1s and Group 2 has one. Every 1 in Group 1 of the K-map sits in the row where A = 0, so those cells include A̅. Those two 1s also sit in adjacent columns that share only B, as the pink arrow shows.
The next common term is B, so Group 1 is the product A̅B. The 1 in Group 2 sits in the row where A = 1. Its column labels are B̅C̅, so that product is AB̅C̅.
Step 5: Obtain Boolean Expression for the Output
OR the group products to form a sum-of-products (SOP) result. That is the overall simplified Boolean expression. For the K-map in Step 4 the simplified output is
Further K-map examples follow.
Maxterm Solution of K Map
A simplified maxterm solution uses the same K-map method as the minterm solution, with the changes listed below.
- Fill K-map cells with 0 for each sum-term of the expression, not with 1.
- Group the 0s, not the 1s.
- Write the Boolean expressions for each group as sum-terms, not as product-terms.
- AND the group sum-terms to obtain the simplified Boolean expression in product-of-sums (POS) form.
