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Simplifying Boolean Expressions using the Laws of Boolean Algebra

  1. May 10, 2010 #1
    EDIT: My apologies if I started this thread in the wrong forum!

    1. The problem statement, all variables and given/known data
    Z = P.Q.R + P.R

    Where
    . = AND
    + = OR
    ~ = NOT


    2. Relevant equations

    (Commutative Law)
    A+B = B+A, A.B = B.A

    (Associate Law)
    (A+B)+C = A+(B+C), (A.B).C = A.(B.C)

    (Distributive Law)
    A(B+C) = AB+AC, A+(B.C) = (A+B).(A.C)

    (Identity Laws)
    A+A = A, A.A = A
    A.B+A~B = A, (A+B).(A+~B) = A

    (Redundancy Laws)
    A+A.B = A, A.(A+B) = A
    0+A = A, 0.A = 0
    1+A = 1, 1.A = A
    ~A+A = 1, ~A.A = 0

    (Demorgan's Theorem)
    ~(A+B) = ~A.~B, ~(A.B) = ~A + ~B

    3. The attempt at a solution

    I don't even know how to go about finding the solution. I'm not recognizing how I should apply the above laws of Boolean Algebra to simplify my problem statement. I'm thinking I must have missed something fundamental while in class. Any tips on how to proceed would be greatly appreciated!

    Just to clarify, I'm not looking to be given the answer, just some tips on how to use the Laws to get to the solution. Thanks in advance! :D
     
    Last edited: May 10, 2010
  2. jcsd
  3. May 10, 2010 #2

    Mark44

    Staff: Mentor

    I take it that you want to simplify Z = PQR + PR

    The two expressions on the right have a common factor. Rewrite the right side in factored form. What do you get?
     
  4. May 10, 2010 #3
    Ahh yes, I am looking to simplify Z = PQR + PR, sorry for not being clear on that!

    After factoring the right side, this is what I`ve come up with.
    Z = Q + PP + PR + RP + RR

    Provided my factoring is correct, I believe I can use the Identity Law AA=A to get -
    Z = Q + P + PR + RP + R

    And then, using the Commutative Law AB=BA to get -
    Z = Q + P + PR + PR + R

    And again using the Identity Law A+A=A to get -
    Z = Q + P + PR + R

    Then using the Redundancy Law A+AB=A to get -
    Z = Q + P + R
     
    Last edited: May 10, 2010
  5. May 10, 2010 #4

    Mark44

    Staff: Mentor

    ??? That's not factored. I have no idea how you got this.
     
  6. May 10, 2010 #5
    Aww crud, I`m sorry. I took the common terms PR, and tried applying the FOIL method (First, Outer, Inner, Last) to it. I thought that`s what factoring was D:.

    I suppose what I need to understand, is what needs to be done, to factor Z=PQR+PR.
     
    Last edited: May 10, 2010
  7. May 10, 2010 #6
    Try Z = PR (1 + Q)
     
  8. May 10, 2010 #7

    Mark44

    Staff: Mentor

    As written, PQR + PR is a sum of two terms. Rewrite this expression as a product of two factors.

    For example 2xy + 2y = 2y(x + 1).
     
  9. May 10, 2010 #8
    Ok, I think I understand now. What I`m doing is, taking the common expression (in the above example it would be 2y) and dividing it out. So I get

    2xy / 2y = x
    2y / 2y = 1

    Which leaves me with 2y(x+1).

    Similarly with the problem I originally posted PQR+PR, I divide PR out of both sides (as it`s the common expression).
    PQR / PR = Q
    PR / PR = 1
    Z=PR(Q+1)
     
  10. May 10, 2010 #9

    Mark44

    Staff: Mentor

    You're really going the long way around to go from Z = PRQ + PR to Z = PR(Q + 1).

    All you need to say is Z = PRQ + PR = PR(Q + 1).

    You should be able to do the dividing in your head - you don't need to write it down.

    Now, what further simplification can you do? There is one identity in your first post in this thread that will be useful.
     
  11. May 10, 2010 #10
    Oh yea, I wouldn`t take the long way like that for my assignment, was just listing out the steps i was taking, so that if I made any mistakes it could be pointed out.

    I`m thinking the relevant identity in this case would be 1+A=1. Using that I can turn
    Z=PR(Q+1)
    into
    Z=PR(1)
    then I can use the inverse of that, 1A=A to get
    Z=PR

    I hope I`ve done it correctly this time, haha.
     
  12. May 10, 2010 #11

    Mark44

    Staff: Mentor

    Right, so Z = PRQ + PR = PR(Q + 1) = PR(1) = PR

    That's not the inverse. 1 is the multiplicative identity. I don't think there's a multiplicative inverse, but the additive inverse of A is ~A, since A + ~A = 1
     
  13. May 10, 2010 #12
    Ahh ok, thank you so much for clearing that up! It`s been driving me batty.

    One more quick question, if y`all dont mind.

    Does factoring always come before simplifications, or are there cases in which the best course of action would be to apply an identify first, and do factoring after? I`m just curious, as in another problem, it seems that it would be best to do just that.
     
  14. May 10, 2010 #13

    Mark44

    Staff: Mentor

    It might be reasonable to apply an identify first. I think you have to take things on a case-by-case basis.
     
  15. May 10, 2010 #14
    I`m thinking it would be applicable in the case of Z=(P+Q).(R+S)+P.(R+~R)
     
  16. May 10, 2010 #15

    Mark44

    Staff: Mentor

    Sure. If there are obvious simplifications, make them.
     
  17. May 10, 2010 #16
    Awesome, thank you again so much, you were a great help to me :)
     
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