Karnaugh Maps

1. Karnaugh Maps By: ShakilNobes

2. History of Karnaugh Maps • The Karnaugh map was invented in 1952 • It was invented by Edward W. Veitch • It was then developed further in 1953 by Maurice Karnaugh

3. What’s the purpose to Karnaugh Maps? • Normally, extensive calculations are required to obtain the minimal expression of a Boolean function • The Karnaugh map reduces the needs for all these calculations • It decides which terms should be combined to obtain the simplest expression. • It Helps to teach about Boolean functions and minimization.

4. What does it look like? W X Y Z

5. How does it work exactly? • Design your truth table • Put your numbers into your Karnaugh Map • Box your 1’s • Create your equation • Draw your diagram

6. I don’t get it???? Here’s an example: Lets say I wanted to take a bunch of LED’s and line them up to make an 8. Ex.

7. Now I want to make it so when I flick on a switch, the light turns onto 1 like this: So If I have 4 switches, and I want to make a 1 appear I flick switch 1. If I want to make 2 appear I have to flick switch 2. But If I want to make 3 appear, I have to flick switch 1 and 2 because 1+2=3, and there’s no switch 3. Another example is there’s no 7 so I have to flick switch 3,2 and 1 because 3+2+1=7.

8. Easy Enough? Well there’s still lots more. So lets say I wanted to design that with all 4 switches and the LED’s. Well first I have to set each set of LED’s to a assigned letter. So the top 3 LED’s are set to A. The middle 3 LED’s are set to D. So on and so forth… A C B D E F G

9. Designing a truth Table • So now we need to design a truth table for each letter. Let’s start with letter A, the top 3 LED’s. So we know that when we the input is zero, the output is zero. So A is included in the number zero because all the lights go around the outside. The only letter that wouldn’t be included in zero is D because D is not lit up when zero is the output or else it would make it the number 8!

10. Truth Table for A OUTPUT INPUT W X Y Z A So to the left here we have a truth Table for A. Because we have 4 Switches, we are going to assign each Input a different letter. W, X, Y, and Z. On the output side, we have 0-9. As you can see were using binary code. 0000 is 0. 0001 is 1. 0010 is 2. 0011 is 3 And so on… So when the output is zero, we know that A is on so we put a 1 there. When the Output is one, we know that A is not lit up So we put a 0. When the output is Two we know A is lit up so we put a 1 and so on… 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 1

11. Final Truth Table for A OUTPUT INPUT W X Y Z A 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 So when the output is one and four, A is both off and for the rest A is all on.

12. The Hard Part Now here is where Karnaugh maps kick in. We now take our truth table and insert it into the Karnaugh map.

13. How does the Karnaugh Map work? W X Well the way we designed the Karnaugh Map, we will read WX, YZ. So we relate back to our truth table. When W and X are both 0, and Y and Z are both zero, the output Is 1. So we fill a one into the box Y Z

14. How does the Karnaugh Map work? W X Well the way we designed the Karnaugh Map, we will read WX, YZ. So we relate back to our truth table. When W and X are both 0, and Y and Z are both zero, the output Is 1. So we fill a one into the box. Y Z 1

15. How does the Karnaugh Map work? W X Well the way we designed the Karnaugh Map, we will read WX, YZ. So we relate back to our truth table. When W and X are both 0, and Y and Z are both zero, the output is 1. So we fill a one into the box. Y Z W X 1 We now go to the next one down, When W is 0, X is 0, Y is 0 and Z is 1, the output is 0. So we fill in a 0 there.

16. How does the Karnaugh Map work? W X Well the way we designed the Karnaugh Map, we will read WX, YZ. So we relate back to our truth table. When W and X are both 0, and Y and Z are both zero, the output is 1. So we fill a one into the box. Y Z W X 1 0 We now go to the next one down, When W is 0, X is 0, Y is 0 and Z is 1, the output is 0. So we fill in a 0 there.

17. How does the Karnaugh Map work? W X We now go and fill in the rest of The boxes. Y Z 1 0

18. How does the Karnaugh Map work? W X We now go and fill in the rest of The boxes. Uh Oh. We ran into a problem. There’s no 1100! So if you don’t have a value in the truth table, you fill it with an x. Like so… Y Z 1 0 0 1 1 1 1 1

19. How does the Karnaugh Map work? W X We now go and fill in the rest of The boxes. Uh Oh. We ran into a problem. There’s no 1100! So if you don’t have a value in the truth table, you fill it with an x. Like so… Now to fill in the rest of the values… Y Z 1 0 x 0 1 1 1 1 1

20. How does the Karnaugh Map work? W X We now go and fill in the rest of The boxes. Uh Oh. We ran into a problem. There’s no 1100! So if you don’t have a value in the truth table, you fill it with an x. Like so… Now to fill in the rest of the values… Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

21. How does the Karnaugh Map work? W X So now that we have all the values put in it’s onto the confusing part! We now have to make a box around all the ones. You cannot box zero’s and here are the restrictions: Boxes cannot be 1 by 1, or 3 by anything. For example 3 by 2 like this… Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

22. How does the Karnaugh Map work? W X So now that we have all the values put in it’s onto the confusing part! We now have to make a box around all the ones. You cannot box zero’s and here are the restrictions: Boxes cannot be 1 by 1, or 3 by anything. For example 3 by 2 like this… NOT ALLOWED! Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

23. How does the Karnaugh Map work? W X So now that we have all the values put in it’s onto the confusing part! We now have to make a box around all the ones. You cannot box zero’s and here are the restrictions: Boxes cannot be 1 by 1, or 3 by anything. You may not make boxes diagonally, but you may match all 4 corners. You may also do this: Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

24. How does the Karnaugh Map work? W X So now that we have all the values put in it’s onto the confusing part! We now have to make a box around all the ones. You cannot box zero’s and here are the restrictions: Boxes cannot be 1 by 1, or 3 by anything. You may not make boxes diagonally, but you may match all 4 corners. You may also do this: Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

25. How does the Karnaugh Map work? W X So now that we have all the values put in it’s onto the confusing part! We now have to make a box around all the ones. You cannot box zero’s and here are the restrictions: Boxes cannot be 1 by 1, or 3 by anything. You may not make boxes diagonally, but you may match all 4 corners. You may also do this. You can join boxes from one side to the other… Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

26. How does the Karnaugh Map work? W X THE MOST IMPORTANT THING: You want to make the biggest box you can, making sure you box all the ones. You can use as many boxes as you want. You also can box a 1 twice. Also, think of x’s as 1’s. You can box them as many times as you want but it is not mandatory to box all the x’s. They are there so you can create bigger boxes with the 1’s. So instead of doing this: Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

27. How does the Karnaugh Map work? W X THE MOST IMPORTANT THING: You want to make the biggest box you can, making sure you box all the ones. You can use as many boxes as you want. You also can box a 1 twice. Also, think of x’s as 1’s. You can box them as many times as you want but it is not mandatory to box all the x’s. They are there so you can create bigger boxes with the 1’s. So instead of doing this: Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

28. How does the Karnaugh Map work? W X THE MOST IMPORTANT THING: You want to make the biggest box you can, making sure you box all the ones. You can use as many boxes as you want. You also can box a 1 twice. Also, think of x’s as 1’s. You can box them as many times as you want but it is not mandatory to box all the x’s. They are there so you can create bigger boxes with the 1’s. So instead of doing this. You want to do this: Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

29. How does the Karnaugh Map work? W X You want to do this because you are trying to create the biggest box you can. The bigger the box, the less work! Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

30. How does the Karnaugh Map work? ANOTHER IMPORTANT THING: Lots and lots of people make boxes that are by 3! Make sure you do not make boxes that are 3 by 1, 3 by 2, 3 by 3 and so on! Now try and get all the ones into boxes! Remember you can use as many boxes as you like! Also you can box x’s and 1’s as many times as you like! W X Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

31. How does the Karnaugh Map work? W X Y Z 1 0 x 1 So I used the 4 corners, a 2 by 2 box in the middle, the 2 by 4 box and the bottom and another 2 by 4 box on the far right side. That includes all the ones, and the biggest boxes I can make! 0 1 x 1 1 1 x x 1 1 x x

32. Creating the formula W X This is what everyone gets confused about at first, but eventually you will get the hang of it! So let’s start with our 4 corners. W is under 0 and 1. Don’t get it? Look: Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

33. Creating the formula W X This is what everyone gets confused about at first, but eventually you will get the hang of it! So let’s start with our 4 corners. W is under 0 and 1. Don’t get it? Look: Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

34. Creating the formula W X This is what everyone gets confused about at first, but eventually you will get the hang of it! So let’s start with our 4 corners. W is under 0 and 1. See how W is represented by the first numbers. So we are not using the 2 middle columns because there’s no boxes going vertically in that row. So for X it would be 0 and 0. Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

35. Creating the formula W X This is what everyone gets confused about at first, but eventually you will get the hang of it! So let’s start with our 4 corners. W is under 0 and 1. See how W is represented by the first numbers. So we are not using the 2 middle columns because there’s no boxes going vertically in that row. So for X it would be 0 and 0. For Y it would be 0 and 1. Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

36. Creating the formula W X This is what everyone gets confused about at first, but eventually you will get the hang of it! So let’s start with our 4 corners. W is under 0 and 1. See how W is represented by the first numbers. So we are not using the 2 middle columns because there’s no boxes going vertically in that row. So for X it would be 0 and 0. For Y it would be 0 and 1. For Z it would be 0 and 0. Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

37. Creating the formula W X Make sense? Well now we create the formula for that box. To do that, we take W and X and we see W is 0 and 1 and X is 0 and 0. X is 0 and 1. Because we have a mismatch with X (0,1) we don’t use it. W is (0,0), so we add a W to our formula. But because the W is zero’s, we label W as W ‘ (W invert). Now we go to our Y and Z. Y is 0 and 1 so we don’t use it. Z is 0 and 0 so we do. It would be labeled Z ‘. Because we are still doing one whole box, we take W’ and Z’ and multiply them together to make W’ Z’ That is our formula so far. Onto the next box. Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

38. How does the Karnaugh Map work? W X Y Z Let’s take the bottom 4 by 2 box. We start with W and X. W is 0,0,1,1. This means that the zero’s and the 1’s cancel each other out so we don’t use W. X is the same thing. 0,0,1,1. The 0’s and 1’s cancel each other out so we don’t use X either. Now onto Y and Z. Y is 1 and 1! This is a match. So now we use Y in our formula. Because Y is two 1’s, it’s just left as Y. You only invert when you have 2 zero’s. Now onto Z. Z is 1 and 0. MISMATCH! So we don’t use Z. 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x So in our formula now we have X ‘ Z’ and Y. Because we have switched to a different box for Y, we put a plus sign in whenever we switch boxes. So our current formula is X ‘ Z’ + Y

39. How does the Karnaugh Map work? W X Let’s move on to our next box. We see that W is 1 and 1 so we use it! X is 1 and 0 so we don’t use it. Y and Z are both 1,1,0,0 so we don’t use both Y and Z. So for this box we only use W, just like in the last 2 by 4 box we only used X. So our current formula is: X ‘ Z ‘ + Y + W Y Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

40. How does the Karnaugh Map work? W X Y Z Now onto our last box! We see that W is 0,1 so we don’t use it. X is 1,1 so we use it. Y is 0,1 so we don’t use it. And Z is 1,1 so we use it! Because we are in the same box still we take the X and Z and put them together to make X Z. So our final formula is X ‘ Z’ + Y + W + X Z 1 0 x 1 0 1 x 1 1 1 x x 1 1 x x

41. Now to take it to logic gates So our current formula is X ‘ Z’ + Y +W + X Z. So let’s start by creating our four switches, and labeling them 1, 2, 4 and 8 (Z would be 1 and Y would be 2 and so on because our truth table tells us so)

42. Now to take it to logic gates So lets start with X ‘ Z ‘. We first have to invert X and Invert Z. So we have to use 2 NOT gates to invert them. So our logic gates will look like this:

43. Now to take it to logic gates We now have to multiply X ’ and Z ’ together to make them X ‘ Z ‘. We do this with an AND gate:

44. Now to take it to logic gates We now have to add our X ‘ Z ’ to Y. We do this with an OR Gate:

45. Now to take it to logic gates We now have to add our X ‘ Z ’ + Y to the W. We do this with another OR Gate:

46. Now to take it to logic gates We now have to add our X ‘ Z ’ + Y + W to X Z. First we have to take X and Z and multiply them together with an AND gate:

47. Now to take it to logic gates Now we have to take our XZ and add it to our X ‘ Z ‘ + Y + W. We do this with a OR Gate.

48. Now to take it to logic gates Now we take the whole circuit we made and attach it to the A part of our LED’s:

49. You have now completed A • Wow that was a lot of work! Now your have to do that for B, C, D, E, F and G! I’ll post my circuit diagrams for you guys to check…

50. Note • My 6’s have a stick at the top (it uses A) my nines have a stick at the bottom (it uses G). Also my 7 only uses A,B and F.