Looping adjacent 1’s for simplification
The expression for output Y can be simplified by properly combining those squares in the K-Map which contain 1s. The process of combining those 1s is called looping.
Pairs – Looping groups of Two 1s
Any adjacent pair of cells marked by a 1 in a K-Map can be combined into one term and one variable is eliminated which is changing i.e. from A to A’ or B’ to B etc.
Any single logical 1 on the map represents AND function. The total expression corresponding to the logical 1s of a map are the OR function (sum) of the various variable terms, which covers all the logical 1 in the map.
F = Σ(2.3) = AB’ + AB = A (B’ + B) = A
Since variable B is changing from B’ to B, it is eliminated right away
Quad – Looping groups of Four 1s
Four cells that are marked as a 1, they can be combined into one term and two variables can be eliminated. A group of four 1s that are horizontal or vertical or form a square in the K-Map is called a Quad.
Octet – Lopping groups of Eight 1s
A group of eight 1s that are adjacent to each other is called an octet. When an octet is looped in a four variable map, 3 of 4 variables are eliminated because only one variable remains unchanged.
There are few tips to remember while simplifying K-Map:
- Biggest decimal number in the given function decides, which K-Map is to be used. For instance, a single variable can define only two decimal values 0 and 1, with maximum value as 1. Two variables can define 22 =4 values, 0, 1, 2 and 3, with maximum value as 3. So if a given function has 4 as the biggest decimal number, it can not be defined by two variables. We need to use 3 variables because by using 3 variables, we can have 23 = 8 decimal values with 7 as the maximum value.
- We should try to cover all 1′s even if they become part of more than 1 loop.
- We look for the biggest loop at first. So if a K-Map has an Octet, it should be circled first, followed by quads if any, followed by pairs if any.
- Pair eliminates 1 variable, Quad eliminates 2 variables and an Octet eliminates 3 variables.
- While looping, we can visualize folding the K-Map like a paper and can loop 1′s present in left most and right most columns of the same row.
- We can also visualize overlapping K-Map in case of 5 and 6 variable K-Maps.
- We can fold and overlap the K-Map only in horizontal and vertical direction but not in diagonal.