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Boolean algebra- cancellation property of addition

  1. Apr 23, 2010 #1
    1. Prove that for all boolean algebras if x+y=x+z and x'+y = x'+z then y=z.

    2. Relevant equations: x+x' = 1, xx'=0, basically we are allowed to use the usual boolean algebra properties.

    3.Attempt: This the second part of a problem, in the first part we had to give and example of why x+y=x+z does not hold for all boolean algebras. Therefore this problem does not allow for cancellation of x on both sides to arrive at y=z. I have tried adding the LHS and RHS of both equations together to get x+x'+y+y = x+ x' +z+z which reduces to 1+y=1+z but I can't subtract the one from both sides in boolean algebra and I can't find a logical reason why they'd cancel. In fact, it seems the next logical reduction is to 1=1.

    I also tried finding and expression for one equation in another and substituting to try and reduce one side to z and the other to x but I all I can come up with is complicated expressions that don't reduce very nicely and on my "best" attempts I've arrived at x+zy=x+yz

    It seems like I am approaching the problem in the wrong way but I'm at a loss for another approach. Any suggestions?
  2. jcsd
  3. Apr 24, 2010 #2


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    Did you try multiplying the two equations together? I didn't actually work it out by hand, but it looks like it should work.
  4. Apr 24, 2010 #3
    Yes, that does appear to work! Thanks so much that never would have occurred to me.
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