Karnaugh Maps - Product of Sum (POS) Expression Reduction (Digital Logic Part 8)

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In this video we will be going over how to map a product of sum Boolean expression on a karnaugh map, how to group variables plotted on the map and how to reduce the pos expression using a karnaugh map.
Note I did another video for sum of product karnaugh maps and some of the materials is repeated in this video.
So what is the purpose of a karnaugh map… a karnaugh map is used to simplify Boolean expressions. It consists of rows of Boolean variable combinations in this case a and b and columns that consists of Boolean variable combinations in this case c and d.
When laying out the map you must only have one of the variable change between rows and columns so you can not place a 11 row next to a 00 row
The combinations are represent as 1 and 0’s. For product of sum karnaugh mapping you have to think backwards. A A is equal to 0 and a A with a line over top is equal to 1.
Now lets look at a product of some formula and put it into the karnaugh map.
The first group aplus b plus c plus d can be maped and represent placing a 0 in the first row first comlum. Because this expression is represented by the first two variables 00 for a and b which is the first row and the second set of variables CD are 00 which is represented by the first column
The second group of variables can be represented by 0010 which corresponds to row 1 column 4.
Completing the same procedure for the rest of the Boolean expression we get the displayed table. You might want to pause here for a minute to take it in. Now great we have a karnaugh map for product of sum of a set of variables now what.
Well in order to reduce this we must under stand how to group variables. You can group variables if they are in adjacent cells within the map or matrix. Here is a diagram that shows all of the possible directions you can group variables.
We can group the values within the cells from side to side and up and down
For the values on the edge of the matrix we can wrap it around to the value on the other side as depicted
All together all directions that we can group look like this…
Now lets do some examples of grouping the first example we have two 0’s next to each other we group them by circling the pair
So now if we were to have another set of zeros next to it would we group it like this???
No we would group it with a box around all values like this… The box around the values trump the side by side only match.
So no if you have a 3 column 2 row set of 0’s would you group it like this???
No in order to group a set of 0’s the adjacent number of 0’s must be even. So being there is 3 0’s in a row we can only group 2 of those 0’s at a time. If there were to be 4 zeros in a row we could group those together. So we would group it with two 2 by 2 boxes like this.
Now lets say we have one 0 all by itself next to a box how do we group this???
We group it with the adjacent cell like so
Now lets say we have 0’s like this do we have a group anywhere???
Yes we do recall that the adjacent cells also wrap around so the grouping would look like this… The ones are grouped by wrapping around to the other side of the map or matrix.
This also applies if there is a box grouping like depicted
Now lets say you have a column of 0’s like this how would you group it like this???
No you would group the 0’s as a single column by 4 row box as depicted.
2 columns of 0’s next to eachother???
You would group it like this with one single 2 column by 4 row box.
So now how would you group this one??? Pause here and think about it…
You would group it like this… with 1 rectangle for the 2 columns and one box that wraps around the the bottom and one box that is at the bottom.
Now following the rules we learned from the previous slides how would we group the original problem…. Pause here to work it out for yourself…
It would look like this… so now you are probably thinking great I know how to group values together what is this used for?
We can use this to reduce to the minimum product of sum expression… the way we do this is by finding the variables that do not change insides each box. then we take the variables that do not change in each box and multiply them to eachother.
So for the box grouping the four corners together the only variables that do not change within this box is b and d. so this box is reduced to b plus d
Now for the blue box that is wrapping around the values in column 1 the only values that do not change are c and d.. So this box can be reduced to c plus d
Now putting this values together we b plus d times c plus d or in boolean algebra terms b or d and c or d