Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

sevenshaded

  • 2 years ago

Find the minterm expansion of f(A, B, C, D) = (A + B + D')(A' + C)(C + D) Please tell me what rules you're applying? I keep getting stuck.

  • This Question is Closed
  1. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Working it out.

  2. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Do you know where to start?

  3. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I tried to take (A' + C)(C + D) and convert it into x+yz form, getting (A'D + C). So altogether it's (A + B + D') (A'D + C). But from there, I'm not sure what to do My my problem is I don't quite know how to have a good strategy for approaching these and I can't see where to apply formulas.

  4. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I'm assuming I need to put it into SOP form.

  5. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Well, they're throwing you a bone here. This is in "product of sum" (POS) form. POS form expresses MAXTERMS. The minterms are everything that is NOT a maxterm. This is the strategy I would use. But here you need to have all variables.

  6. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I wouldn't put it into SOP form at all, actually. I'd get the maxterms, then just ...the minterms are everything the maxterms are not.

  7. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So then, would I convert these into binary form, find the numbers for the minterm and then...?

  8. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Well, first the products are incomplete. All three products need to have A, B, C, and D in some form. So you can use the fact that XX' = 0 to add them in. Because when it comes out, it's as though they were never there.

  9. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So for example, with the first product (A + B + D'), you're missing "C". So: (A + B + CC' + D')

  10. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Now distribute (A + B + C + D')(A + B+ C' + D') See?

  11. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Oh, yeah, I forgot that step. But I mean after that, would I just find the actual numbers for the minterm by using the numbers that aren't there for the maxterm?

  12. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yea, cancel out redundant maxterms, and express them as a decimal number. Those are your maxterms.

  13. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Then everything not present in the maxterms is a minterm.

  14. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Wouldn't I still have to simplify them algebraically? The numbers I mean. When I convert them back into variables?

  15. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Well, if you look at the first product only: (A + B + C + D')(A + B+ C' + D') You have 1110 and 1100, this is 14 and 12. So if there were NO other products your max terms would be: \[\Pi M(12, 14)\] Minterms would then be everything else.

  16. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Of course, you have two other products that are going to distribute into more products and some of them will cancel, but I don't see any crazy boolean algebra that you have to do here at all.

  17. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ah okay. I'll work on it then.

  18. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    And actually, that's not correct in my example. With maxterms, 1 is actually 0, so you have: 0001 and 0011, giving 1 and 3

  19. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So minterms would be everything 0 - 15 other than 1 and 3

  20. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yep.

  21. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I see. I'll go ahead and jump on it. Thanks for the help. :)

  22. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    No prob, let me know what you get there.

  23. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So far, I multiplied (A + B + CC' + D')(A' + BB' + C + DD')(AA' + BB' + C + D) And that resulted in AC + ACD + A'BC + A'BD + BC + BCD + A'CD' + CD' and now I'm looking for consensus terms or something to get rid of. I want to use BC + BCD to get rid of BC, but I'm not sure if that's legal?

  24. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Overcomplicating.

  25. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Throw that out the window.

  26. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Lol okay

  27. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    First term: (A + B + D') = (A + B + CC' + D') = (A + B + C + D')(A + B + C' + D') =0001 0011 Second term: (A' + C) = (A' + BB' + C + DD') = (A' + B + C + D)(A' + B + C + D')(A' + B' + C + D)(A' + B' + C + D') =1000 1001 1100 1101 Third Term: (C + D) = (AA' + BB' + C + D) = (A + B + C + D)(A + B' + C + D)(A' + B + C +D)(A' + B' + C + D) =0000 0100 1000 1100 Every single one of those binary numbers there is a maxterm. Some of them repeat, and you can get rid of the redundant ones.

  28. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Looking at the first term, for example, do you see how the CC' distributes? We have to account for both the C and the C'. That gives what we ORIGINALLY had (A + B + D'), but now we have two variants. One with a C and one with a C'. Thats why we have (A + B + C + D')(A + B + C' + D')

  29. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yeah, I see. I didn't need to distribute the way I did.

  30. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Nope, not at all, because its still in POS form. Distributing just makes it needlessly complicated. Finding maxterms and minterms is a first step in SIMPLIFYING a switching circuit to its simplest possible form.

  31. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Using boolean logic is brute force and may not result in a completely simplified circuit. This method is a first step in two methods that will ALWAYS give you the simplest switching circuit.

  32. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Is it ever the case when SOP form is considered simplified? I have no idea why I interpret "simplify" to mean "put it either in the smallest POS or SOP form"

  33. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I mean. Switch those around.

  34. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Well, ultimately these turn into a bunch of and/or gates and inverters and so forth. The simplest expression is the one that uses the least amount of them really. So you have the smallest number of or's, the smallest number of and's....and so forth.

  35. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Oh, I see.

  36. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So yes, either POS form can be simplified, or SOP can be. Doesn't matter. In this case there are an equal number of minterms to maxterms. But this does not mean much......yet

  37. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    One more thing. Just in general, when simplifying boolean algebra overall, do you have any tips for how to approach problems that ask for simplifying something into POS or SOP respectively? It's just really hard for me to see, for example, when to call something X and where to group something into maybe Z or Z' or something. So I'm practicing, but... Yeah. Lol.

  38. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I start out to simply and end up complicating. Or start out trying to do POS and end up doing SOP. Stuff like that. I try to stick to the simplification theorems for each respectively but I notice there's times when using another method for half the problem is advantageous.

  39. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Well, the whole thing is when dealing w/ minterms and maxterms, the final expressions must ALL have ALL elements of the function. To introduce the missing elements, when using maxterms (POS) you have to introduce them like they were never there. They have to be transparent. Thats why with maxterms, we use XX'. Because XX' = 0. So or'ing an expression with 0 has no effect. See? When using minterms, we use X + X'. Because X + X' = 1. So when you "and" an expression with 1, it has no effect. See?

  40. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Right, I see.

  41. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So an SOP example might be, find the minterms of: F(A, B, C) = A'B' + AC Well, there's two expressions there. The first one is missing C, the second is missing B. So we say: A'B'(C + C') + AC(B + B') We get A'B'C + A'B'C' + ABC + AB'C = 001 + 000 + 111 + 101 So the minterms are 0, 1, 5, 7

  42. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    And there's no need to attempt anything like eliminating consensus terms, etc. in these sorts of problems.

  43. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Just use the number values.

  44. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    No, I mean...in this case you're already given something in POS form, which is why I say they were throwing you a bone. If you're given something that's a mix, you may have to use some boolean trickery to get it into either form.

  45. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    But once its in POS or SOP form, stop trying to simplify and start using what was used here.

  46. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Okay, that makes sense.

  47. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Very good. Digital circuits, I presume?

  48. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I'm not sure what the formal name of the class is. It's CpE 100 here. I'm a CS major, so this is pretty much my last class in this, lol.

  49. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Ah, then Digital Logic or something like that.

  50. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Computer Logic Design I There we go.

  51. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Sounds about right. Good luck and enjoy :]

  52. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Lol, thanks. You helped a lot! :)

  53. qpHalcy0n
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    The maxterms are 0, 1, 3, 4, 8, 9, 12, 13 The minterms are 2, 5, 6, 7, 10, 11, 14, 15 btw ;]

  54. sevenshaded
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Haha, thank you. :P

  55. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.