KMAP3 is a very powerful Boolean calculator is a very powerful tool for evaluating, calculating  of Boolean Algebra expressions. The best Boolean calculator on internet ! Even wolframalpha.com does not have such powerful tool. This software  is a very useful tool for testing  DeMorgan theorems and investigating set theory identities. In addition to handling of Boolean algebra calculation it can also handle more basic Karnaugh Map(KMAP) problems, in which better optimization are done. This Boolean calculator can handle terms with 4 up to 16 variables. No of vars can be controlled by the up/dn switch box. Instead of Venn diagram a KMAP diagram is used for better visualization of the terms and expressions. This app is also very useful for digital electronic engineers to optimize their digital circuitry. Some previous knowledge of Boolean Algebra and variable name format readers are assumed to have.     In the opening screen you will see this form

 

Basic Operation.

 This calculator has 2 input editors X and Y and an output editor Z . The Z edit box is updated when corresponding Boolean operations AND, OR, NOT icons are clicked. B1(buffer 1) and B2(buffer 2) edit boxes are simply 2 temporary buffers to store the intermediate results. The OZ (Optimized Z) is just to notify you that the value in Z edit box can be further optimized and reduced to shorter expressions . Using drag and drop values of edit boxes X, Y, Z, B1 and B2 and OZ can be copied into another edit box without using copy and paste. Drag & Drop click on the yellow labels.

 

 In case of  NOT(complement)  if you don’t want the NOT value overwrite the Z input, you can type the Boolean in the Y edit box , but you must enable auto calculation by clicking on its menu. Then as you type your Boolean expression in the Y input the Z input automatically updated with the complement value(NOT) of  Y input. Notice when NOT icon is clicked the X edit box is automatically disabled and you can not type in it, unless you click on another Boolean icon.

 

 Important in the bottom part of screen you see a grid which is automatically updated with the Karnaugh Map boolean expression in the active edit box(edit box that has the focus) as you type in to give you a visual feeling how the KMAP looks like. As it was mentioned KMAP grid is better boolean representation than Venn Diagram.

Some examples

In these example lo state of logic is presented lower case characters for example instead of A' we use a

For all these for examples we assume X=Ab+Bc  and Y=aB+bC

1-Boolean AND operation. Click on the AND icon. You will see Z input is Z=AbC+aBc . If in the main menu you select Autoà calculation then you don’t need to click on the AND icon. As you type the Z output box is automatically updated with logical AND  value of X AND Y .  You can think of Z as the raw out put while OZ is optimized value of Z. If you want a NAND operation click on NOT and you get Z=AB+Ac+BC+aC+ab+bc

 

2-Boolean OR operation. Click on OR icon . The new value of  Z=Ab+Bc+aB+bC. If you want the NOR operation then you click on NOT and you get Z=ABC+abc. Notice in the case of OR the most optimized answer is Z=Ab+Bc+aC. Which is obtained by using more comprehensive Karnaugh Map(KMAP) method of simplification and optimization. To use this advanced optimization just Click on BooleanàZ input to kmap grid and then click on the Simplify in that form main menu.

 

3-Boolean XOR operation . Click on XOR icon.  The new value of Z=ABc+Abc+aBC+abC  but in the optimized edit box(OZ) you will see value of Ac+aC which is the same as A XOR C. If you want XNOR instead just click on NOT icon again and you will see result AC+ac . But in this case the optimized answer was calculated in the first run.

 

Now lets look at different menu items

Commands available from Boolean menu . The last menu item will allow you to save the visual image of current edit box Karnaugh Map (KMAP) representation. Later we will talk about the 4th item in the menu Copy Z edit to KMAP grid . Which will invoke the KMAP editor, that will allow you to do a better optimization and enter the data by clicking on KMAP grid instead of typing it. Clicking on this menu the same effect as ShowKmap but the data will be copied.

 

 

Commands available from Check menu to check for  Boolean syntax error.

 

 

Auto Calculation will calculate the Z input as you type in the X or Y inputs without needing to click on any icon. If you want this handy feature make sure the right icon was clicked on first.

 

 

By clicking the Projects menu simply a new form will popup that have the boolean equations of a 4 bit adder. Here you will see one complex project of 8 input . This page is very experimental and still under construction and consideration. It is mainly for information only.

 

 

 

The first menu item ShowKmap is a very important part of this application. The ShowKmap menu is very similar to Copy Z edit to KMAP grid  that was mentioned in the boolean menu . When you click on it this form will popup. This form using File menu will allow you to save the KMAP data as both text and binary file .

In the left you see an editor box in which each line can only contain one term only. You can not type + sign in it. And in the right you see the KMAP visual presentation of  the terms in the left hand editor. Unlike the editor in the calculator form you will see true visual presentation of low logic by seeing an upper case character with a bar above it. To see how this form works lets go back to the 2nd example using OR operation. We have X=Ab+Bc , Y=aB+bC the Z=Ab+Bc+aB+bC . But we are not sure if Z is really optimized. By clicking on the Copy Z edit to KMAP grid  you will see this populated form

The menus of this form are

All the features and commands are available from the main menu, which are self explanatory.  The 4th item in the main meu is HiLite . By clcking on this menu the terms that are checked in the left editor are temporary flashed yellow for better visual understanding and distinguishing and overlaping that boolean terms can have.

The terms Ab, Bc, aB, bC  are displayed in the left editor with correct boolean visual presentation. And in the right side the KMAP visual representation . In this example 6 variables from A ..F are used but we could have simply just use 4 variables from A to D using the vars switch . In-order to simplify and optimize this KMAP to the max now you click on the Simplify menu. You will see  the new screen bellow

 

 

This is the screen where unlike calculator which uses string comparison and reduction method used a binary brute force calculations are done to find the minimized number of terms. We notice 2 more new terms Ac and aC are also added because they are valid terms that will cover the cells of KMAP. Here you can do some manual tweaking to see which terms can be removed or click of Show Optimized to see which one of these 6 terms can be removed. If you click on Show Optimized  you will see the same screen with suggested terms that can be removed.

 

 

The software will suggest to you that 2nd(Ac) , 4th(aB) and 6th(bC) terms can be removed. Next you click on remove . This KMAP is now optimized to Ab+Bc+aC . By clicking on the Action menu you see selection of choices with where and how save and copy these 3 optimized terms.

In the 2nd example given X=Ab+Bc+Cd+De+Ef  and (1)Y=aB+bC+cD+dE+eF find  Z=X  XOR Y . The Z output of this XOR  is   Z=ABCDEf+ABCDef+ABCdef+ABcdef+Abcdef+aBCDEF+abCDEF+abcDEF+abcdEF+abcdeF and OZ=ABCDf+ABCef+ABdef+Acdef+aCDEF+abDEF+abcEF+abcdF . Z has 10 and OZ has 8 terms . We Click on the menu Boolean-àCopy Z edit to kmap grid . We will see the KMAP for with populated date

 

 

Next we will click on the Simplify menu and we will see a form that has a new optimized solution with 8 terms

 

 

Min and Max column indicates how many times that blobk of variable is chared with other variables. If Min is 1 then that term can not be removed. But if Min grater than 1 it is possible to remove it if KMAP blocks of those terms cover all of it blocks. We now click on Show Optimized button and we will see. 6th and 8th terms can be removed. Click on Remove button

 

from Action menu you can save this data or copy it back to calculator for further operation.

 

 

Z=ABCDf+ABCef+Acdef+aCDEF+abDEF+abcdF has only 6 terms. To double check this calculation we can use the Boolean identity  X= (X xorY) xor Y  where X=ABCDf+ABCef+Acdef+aCDEF+abDEF+abcdF  and Y=Ab+Bc+Cd+De+Ef  . X xor Y should give us  Z=aB+bC+cD+dE+eF . But we will notice Z=AbC+AbD+AbE+AbF+BcD+BcE+BcF+CdE+CdF+DeF+aBc+aCDEF+aCd+aDe+aEf+abDEF+abcdF+bCd+bDe+bEf+cDe+cEf+dEf  and  OZ=aB+aC+aD+aE+bC+bD+bE+bF+cD+cE+cF+dE+dF+eF , although both look very different than aB+bC+cD+dE+eF  but indeed they are the same. With the same method we copy the OZ data into  the KMAP form and we will see it populated

 

 

 

 

 

Next we click on Show Optimized

 

As we the final optimization result is

         aB+bC+cD+dE+eF the same as (1)Y=aB+bC+cD+dE+eF

 

 

If we copy the Z data in the KMAP form, we will see the a different populated form with different terms, but same kmap grid.

By repeating the same above procedure we will get the same final simplification and optimization.

         aB+bC+cD+dE+eF the same as (1)Y=aB+bC+cD+dE+eF

 

 

 

 

 

 

 

 

 

Using the Boolean calculator  it is easy to establish these identities

using set theory notation    ∪=OR    ∩=AND   =XOR

in the examples sometimes we use XOR abbreviation, sometimes   symbol

 

X=a∪b∪c∪d    is equivalent to  boolean notation X=a+b+c+d      equi     X=a OR b OR c OR d

Y=A∪B∪C∪D  is equivalent to boolean notation X=A+B+C+D  equi      X=A OR B OR C OR D

……

X=a∩b∩c∩d    is equivalent to   X=abcd     equivalent    X=a AND b AND c AND d

Y=A∩B∩C∩D  is equivalent to  X=ABCD  equivalent   X=A AND B AND C AND D

 

Example 1

Assume X=a+b+c+d   and Y=A+B+C+D

X AND Y=Ab+Ac+Ad+Bc+Bd+Cd+aB+aC+aD+bC+bD+Cd using basic optimization

X AND Y=Ab+Bc+Cd+aD+Ac   A better optimization using KMAP form

X  NAND  Y=ABCD+abcd

X OR Y =1

X XOR Y =ABCD+abcd

NOT(X  XOR Y)=Ab+Bc+Cd+aD+Ac

(Ab+Bc+Cd+aD+Ac) XOR (a+b+c+d)=ABCD

(Ab+Bc+Cd+aD+Ac) XOR (A+B+C+D)=abcd

 

Notice in this example   X  XOR  Y= X  NAND Y

……………………………………………………….

Example 2

Assume X=a+bc+d   and   Y=A+BC+D

X AND Y =Abc+Ad+BCd+aBC+aD+bcD  using basic optimization

X AND Y =Abc+Ad+BCd+aD                    A better optimization using KMAP form

X  NAND  Y=ABD+ACD+abd+acd

X OR Y =1

X  XOR  Y =ABD+ACD+abd+acd

 

Notice also  in this example   X  XOR  Y= X  NAND Y

………………………………………………………..

Example 3

Assume  X=abc+bcd+acd+abd and   Y=ABC+BCD+ACD+ABD

X  AND Y=0

X  OR  Y=ABC+ABD+ACD+BCD+abc+abd+acd+bcd

X  NOR Y=ABcd+AbCd+AbcD+aBCd+aBcD+abCD

X  XOR Y=ABC+ABD+ACD+BCD+abc+abd+acd+bcd

 

Notice   in this example      X OR Y=X  XOR  Y  -à     X+Y=XY

………………………………………………………..

Example 4

Assume  X=ab+ac+ad+bc+bd+cd and   Y=AB+AC+AD+BC+BD+CD

X  AND  Y=ABcd+AbCd+AbcD+aBCd+aBcD+abCD

X  NAND Y=ABC+ABD+ACD+BCD+abc+abd+acd+bcd

X  OR  Y =1    X  NOR  Y =0

X  XOR  Y=ABC+ABD+ACD+BCD+abc+abd+acd+bcd

 

Notice in this example  X AND Y=X  XOR Y  -à        XY=XY

 

 

 

ExclusiveOR  XOR=  has very interesting properties that we can check with Boolean Calculator

 

(A  B)(CD)=ABCd+ABcD+AbCD+Abcd+aBCD+aBcd+abCd+abcD

(A  B)A=B

(A  B)B=A

A  b= NOT(A  B)

(A  B)=(a  b)

(A  B)CD=ABCD+Abc+Abd+aBc+aBd+abCD

(A  B)(C+D)=ABC+ABD+Abcd+aBcd+abC+abD

(A  B)(A+B)=AB  

(A  B)AB=A+B

(ABCD+abcd)(A+B+C+D)=a+b+c+d

(ABCD+abcd)(a+b+c+d )=A+B+C+D

ABCD  abcd=ABCD+abcd

 

If you have any questions or comments and report a bug please contact motahed1@yahoo.com

Kmap3 is a Freeware program - source code not available

 For downloading the program Right Click on the link bellow and chose SaveAs . After download finished rename it to KMAP3.EXE

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