Simplifying Boolean Expression using K Map
Minterm Solution of K MapThe following are the steps to obtain simplified minterm solution using K-map. Step 1: Initiate Express the given expression in its canonical form Step 2: Populate the K-map Enter the value of 'one' for each product-term into the K-map cell, while filling others with zeros. Step 3: Form Groups
- Consider the consecutive 'ones' in the K-map cells and group them (green boxes).
- Each group should contain the largest number of 'ones' and no blank cell.
- The number of 'ones' in a group must be a power of 2 i.e. a group can contain
- Grouping has to be carried-on in decreasing order meaning, one has to try to group for 8 (octet) first, then for 4 (quad), followed by 2 and lastly for 1 (isolated 'ones').
- Grouping is to done either horizontally or vertically or interms of squares or rectangles. Diagonal grouping of 'ones' is not permitted.
- The same element(s) may repeat in multiple groups only if this increases the size of the group.
- The elements around the edges of the table are considered to be adjacent and can be grouped together.
- Don’t care conditions are to be considered only if they aid in increasing the group-size (else neglected).
Hence the next term is B. This yields the product term corresponding to this group as A̅B. Similarly the 'one' in the Group 2 of the K-map is present in the row for which A = 1. Further the variables corresponding to its column are B̅C̅. Thus one gets the overall product-term for this group as AB̅C̅. Step 5: Obtain Boolean Expression for the Output The product-terms obtained for individual groups are to be combined to form sum-of-product (SOP) form which yields the overall simplified Boolean expression. This means that for the K-map shown in Step 4, the overall simplified output expression is A few more examples elaborating K-map simplification process are shown below.
Maxterm Solution of K MapThe method to be followed in order to obtain simplified maxterm solution using K-map is similar to that for minterm solution except minor changes listed below.
- K-map cells are to be populated by 'zeros' for each sum-term of the expression instead of 'ones'.
- Grouping is to be carried-on for 'zeros' and not for 'ones'.
- Boolean expressions for each group are to be expressed as sum-terms and not as product-terms.
- Sum-terms of all individual groups are to be combined to obtain the overall simplified Boolean expression in product-of-sums (POS) form.