# How to Reduce the expression using Karnaugh map. Implement using universal gates. 1) f= ∑(0,2,3,5,6,7,8,9,11,13,15) 2) f= ∑(2,3,5,6,7,9,11,13,14) 1) f=...

How to Reduce the expression using Karnaugh map. Implement using universal gates.

1) f= ∑(0,2,3,5,6,7,8,9,11,13,15)

2) f= ∑(2,3,5,6,7,9,11,13,14)

1) f= ∑(0,2,3,5,6,7,8,9,11,13,15)

2) f= ∑(2,3,5,6,7,9,11,13,14)

*print*Print*list*Cite

### 1 Answer

To make the Karnaugh map, we simply convert all of the true conditions to binary. Showing the map will be difficult in this text-based system...but we'll try!

Let's say the horizontal part of the table is AB, where A and B are the most significant digits of the binary number. The vertical part will be CD, where C and D are the least significant digits. For example, given a number 10 (0110 in binary), A=0, B=1, C=1, D=0.

Instead of a truth table which we can't make, we'll use the plot tool to make Green and Red dots, where Green will be "True" and Red will be "False." From left to right, recall that AB will go in this sequence: 00, 01, 11, 10. Similarly, from top to bottom, CD will go from: 00, 01, 11, 10.

First K-Map: F= `sum` (0,2,3,5,6,7,8,9,11,13,15)

Now, to solve using K-maps, we simply start making groups. Keep in mind, we can wrap the grouping around the edges the way the map is designed.

Using groupings, we can group squares of four on the bottom left, middle, and right middle and we can group two points on the top corners (It may help to draw the K-Map yourself...)

The resulting function is as follows:

`F=barBbarCbarD + barAC + AD + BD`

The second K-Map is as follows:

F = `sum` (2,3,5,6,7,9,11,13,14)

Based on grouping the square on the bottom left, and a bunch of 2-square groups, we get the following equation:

`F = barAC + AbarBD + BCbarD + BbarCD`

You may be able to reduce further using common gates and some boolean algebra (for example, `BCbarD + BbarCD = B(CoxD)` where `ox` is XOR), but the above relations are the most-reduced forms that we'll get from the Karnaugh maps of the functions outlined.

Hope that helps, and I'm sorry I couldn't find a better representation for the K-map!

**Sources:**